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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?

Emily
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Charles Khunt
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  • What kinds of questions are you talking about? I think a lot of it does come down to recognizing certain tricks and patterns, and you build up this ability with experience. How often do people truly think outside of the box? – wj32 Nov 01 '12 at 20:57
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    "What can I do to improve my situation?" (1) Do A LOT of problems. (2) Read George Polya's "How To Solve It" – BobaFret Nov 01 '12 at 21:05
  • Hmm not exactly sure how to answer this. Just questions in general on any topic say textbook practice problems or questions in: http://projecteuler.net/problems although these are more math puzzle types. – Charles Khunt Nov 01 '12 at 21:06
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    I think that how we perceive ourselves, specifically how we perceive ourselves in terms of "what I'm good at" or "what I'm bad at" can be self-fulfilling. I think one's attitude when encountering novel situations, in general, like new problems, has a lot to do with how successful one is in handling the situation: if one develops confidence in one's competence, one is more likely to persevere. One can be fearful, intimidated (retreat); one can feel challenged and stimulated; etc... – amWhy Nov 01 '12 at 21:07
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    I added a couple tags; hopefully, these tags will counter the "not constructive" close vote. – Emily Nov 01 '12 at 21:08
  • A lot of difficult textbook problems (at a proof-based level) simply take a lot of time, more than anything else. As amWhy said, you need to persevere. Don't expect to finish a problem in an hour, a day, or even a week. – wj32 Nov 01 '12 at 21:17
  • The Art and Craft of Problem Solving is a great book. – littleO Nov 01 '12 at 21:44
  • The three steps you gave is how I would say I and most people I know do math. But after solving a problem, you have to pick at it and see why it worked. It's the same idea of "recognize small changes", but a more attentive person sees more ideas in a single problem. After many problems those "small changes" naturally compound into the kind of thinking you're looking for. And the next step should be something like "try random manipulations even if you don't see a solution in them, just to see how things work", rather than "blank". It's like being an artist: your worst enemy is the blank page. – Robert Mastragostino Nov 01 '12 at 23:29
  • The information is much too vague to diagnose your problem. I suggest that you take an actual problem where you failed to find the solution yourself after a long time, but understood the solution and describe where you were blocked and what you were thinking when you were blocked. Ideally, you do this in real life with an expert. Less ideally, you do it here. There are plenty of reasons why one might be stuck during problem solving. For example, you might not be looking at problems of the appropriate level, or you might not have any concept of playing with a problem instead of solving it. – Phira Nov 01 '12 at 23:35
  • Does this answer your question? How do I get good at Math? – user1110341 Oct 01 '23 at 06:50

6 Answers6

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

Rudy the Reindeer
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    Hear hear! A terrific book: we've based part of a course on reading and doing maths for our first year undergrads on it. It helps them not just with problem solving, but also with understanding what it is we do when we do maths. – user12477 Nov 01 '12 at 21:15
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    @amWhy That's a different book. See this instead. – Michael Greinecker Nov 01 '12 at 21:24
  • @Michael: thanks for checking that out and pointing it out!...oops, seems I've posted an incorrect link! I'll delete it at once! – amWhy Nov 01 '12 at 21:38
  • Dear @amWhy, I read your now deleted comment as an alternative recommendation. – Rudy the Reindeer Nov 02 '12 at 07:34
  • @MattN Thanks for your comment; I'll "repost" the link here as a different book, perhaps worth looking into. – amWhy Nov 02 '12 at 13:19
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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!

Rankeya
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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.

glebovg
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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.

Makoto Kato
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I got a PhD and a postdoc in pure mathematics and I just can talk from my experience. I think that mathematical thinking can be improved with your experience solving problems and reading. For me, there are 2 options.

Option 1: improve your mathematical thinking by yourself. This means trying to approach the problem from all the possible points of view that you can imagine. Organize them, try to apply them one by one and draw a lot. Try to improvise and start solving similar problems in simpler versions. This is very hard to do, especially when you learn in a systematic way because this requires creativity but if you spend time doing this even if you don't solve it you can grow a lot and develop intuition. Warning! Do not spend more than 1 week with the same problem. Not all people solve problems quickly and that is just fine. If you cannot solve a problem after your hard work is a good idea to ask for help (books, mentors, the internet, etc) or just leave it in a special list and move forward. You will be able to solve that list in the future. Reading some comments reminds me that teaching to others the exercises that you can solve is a very powerful way to improve your mathematical thinking, it helps to organize your knowledge and discipline your mind. Please be patient with others. Some day someone will be patient with you and you will need it!!!

Option 2: improve your mathematical thinking using help. Some people may think that asking for help from others or books destroys your creativity and limit your mathematical thinking to the creativity of others. But only a few gifted can afford that. I think that there is nothing wrong with gathering some strategies of others to enrich your own bunch of tools. Consult a friend, professor, books or forums like this. When you ask for help the method to solve a problem is something that you haven't thought of, however, the new experience can help you to solve new problems in the future. When you are facing a new kind of problem and you do not have a clue where to start, look for examples and solved exercises. If you are in high school most likely there is a lot of reading material, examples and solved exercises for the topics that you are interested in.

In the end, your experience solving problems and your background will develop your mathematical thinking, and you can do it using options 1 and 2.

Chilote
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Old thread, but I came across this and wanted to pitch in my 2 cents.

I remember when I first got to college and was studying mechanical engineering. My high school education taught me the plug-n-chug method of thinking, so topics like differential equations, physics, let alone, linear algebra, dynamics, thermo, mechanics, etc. were really really difficult for me.

Somehow I struggled through it though, and graduated, but I always felt uneasy about having as solid of problem-solving skills in my educational foundation as I wanted on it. Especially since I was now working (tho my day-to-day work didn't require those specific skills). I ended up making a hack solution and practiced one math or physics problem a day on my own. I felt like I really came to understand those things since now I took the time to go through them myself, and see where all the formulas were derived from. Knowing that, I knew better when I could apply an equation, and in what manner.

I actually came across this site later: www.learnerds.com which pretty much was what I was looking for. An interesting (semi-realistic) math/engineering/science question a day with a good solution, and the authors are great at responding back to your comments, regardless of your level.

Ryan
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