I generally think solving a maths problem is one part experience, one part skill, and one part luck. So let's take these one-by-one.
Experience:
This is the most obvious one and other people will no doubt mention it so I won't belabour the point. The more problems you solve, the better you will get. But there are a few caveats:
- Don't just solve problems you can easily do already. Solving a million quadratic equations is not going to improve your analysis. Splitting a million expressions into partial fractions will not improve your group theory. Using the cosine rule a million times will not improve your calculus. I’m sure you get the point. Getting the groundwork is important, and it's crucial to be able to do the basics before you put them in a problem solving context, but you can't just do this.
- On the other hand, there’s no use attempting problems you can’t even begin to understand. If you’re very new to calculus, looking at second order differential equations isn’t going to help you differentiate polynomials. If you don’t have a clue how to do any of the questions on an exam, you should move to an easier one. In my experience, you learn the most when you can do 60–70% of the problems you attempt. That way, there is plenty of opportunity to learn new techniques, but you won’t get discouraged, and often finding the answer yourself helps the relevant technique stick in your mind. Everyone remembers some difficult problem they finally cracked, but almost no one remembers a difficult problem they looked up.
- Persevere with the questions you’re trying. If you’ve only been working for a few minutes, there’s absolutely no use looking to the answers. For me personally, I keep grappling with a problem until I reach a stage where in the last 20–30 minutes I’ve made absolutely no progress. At this point, I briefly glance at the solution to get some kind of hint, then keep working myself. The longer you work on a problem, the more you benefit from knowing its solution.
- You must actually look at the solutions. You can still derive some benefit from answering questions and ignoring those you get wrong, but you’ll benefit so much more if you read the solutions afterwards (even if you already solved it!). And when I say read, I mean take the time to carefully work through the solution given, making sure you understand every step, and making sure you know where you went wrong.
So the more experience you have, the more you will have trained yourself to think critically about problems. But it goes deeper than that. Eventually you’ll start to recognise problems, or at least see similarities between new problems and ones you’ve already solved. You’ll develop much better intuition and have a larger arsenal of attack.
Skill:
A lot of people think of skill as something you’re born with — you either have it or you don’t. To a certain extent, I agree, some people are more mathematically inclined than others, but at the same time any given person can improve enormously with the right training.
A large part of skill comes from experience. The more maths you do, the quicker you’ll be, and you’ll be more accurate too. But there’s an equally significant portion which comes from the right mindset. There are entire books dedicated to the subject (I’m looking at you, How to Solve It) but briefly:
- You need to be able to examine a problem and reduce it into pure information. If you’re doing a set of structured questions, there will very rarely be irrelevant information given to you. So look at what the question is telling you, think about what it means, and what it implies. Think how the various pieces of information link together and keep repeating this process to make sure you bleed the information for everything it has. Of course, if you’re doing original mathematical research, the state of play is quite different. You have no idea what information will be helpful and what will be redundant. However, I find the best tactic is to keep a separate sheet of paper where you jot down any intermediary results you’ve proven, regardless of whether you think they’re helpful, and keep looking at it for inspiration.
- You should be able to break a problem down into smaller problems. If you can’t solve something, think of a similar, but simpler, problem and solve that. Maybe you could solve the case , or solve it when is prime, or solve it when the quadrilateral is a square, or one of the variables is fixed, or whatever. Try to relate the techniques used in the easier cases (or indeed the result itself) to the case in hand.
- Finally, you should be able to check your answer. I don’t just mean going through the algebra, though that’s certainly useful, but ask yourself whether the answer makes sense contextually. If you’ve calculated that a person walks at 100 mph, or they weigh a million tonnes, then you’re almost certainly wrong. But those are pretty obvious examples; often it can be more subtle. Is your equation symmetrical about the variables and should it be? What happens as a value goes to infinity, or zero? Has your working created redundant solutions?
Luck:
This is the part over which you have the least control, but of course experience still plays an important role here. Often what we call luck is actually just a super refined intuition. There is a common expression that says “the more I practice the luckier I get” and I think it’s absolutely on the nose.
However, there is always going to be an element of luck. Maybe you had a random flash of insight that everyone else missed, or maybe you just could not see one crucial piece of information. Maybe you made some unfortunate mistakes or starting going down the wrong path for a long time.
Your ability to solve problems will vary a lot with mood, tiredness, health, random variation, etc. Sometimes the best tactic to solve a problem is to leave it for a few hours — or maybe a few days — then come back to it. Obviously this isn’t applicable to an exam, but that’s just life for you.
I realise I’ve written an essay here but I hope it’s useful. Good luck!
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