Dear Uncle Colin,

How can I get better at the problem-solving side of maths?

- Probably Really Obvious But Looks Extremely Mysterious

Hi, PROBLEM, and thanks for your message!

The generally excellent @drmackiver has a simple process for solving problems: see what other people with similar problems have done, and try that ((I may be paraphrasing slightly.))

That’s very good advice if your goal is to solve the problem in front of you, but (slightly) less soo if your goal is to get better at solving problems generally.

I have a slightly longer process you’re welcome to borrow.

1: What, exactly, is your problem?

The first step in solving a problem is to figure out exactly what the problem is. In a textbook or exam question, you’ll want to make sure you’ve understood all of the key information, and ignored anything irrelevant. If you’re doing maths in the wild, you’ll need to do the same thing – only it’s unlikely to be set out in front of you in quite the same way.

2: What should the answer look like?

This is a question with several aspects to it. On one level, it’s “what type of answer do you need?” Are you looking for a number? A graph? A function? Are there units attached to it?

On another, it’s “how will you know whether your answer is correct?” What tests could you do after you have the answer to decide?

3: What have you tried already? What held you up?

It might be that you just don’t know where to start. That’s fine. We’ll get to that. But if you’ve had a stab at it and got nowhere, it’s worth articulating what it was that you tried, and where it seems to have gone wrong. Were there any different steps you could have taken? Are there any places you could reasonably have gone wrong? ((Having a commonplace book in which you write down mistakes you commonly make and want to avoid is a godsend here.))

4: What if you don’t know where to start?

Does it look like something you’re familiar with? Have you done similar problems in the past? What approaches worked for them? What do you know about such problems?

Generate as many possible avenues to explore as possible. The problem is made up of different elements. Which ones could you tackle first?

Can you make a plan of how you expect the solution to pan out? How would someone you think is smarter than you approach it?

5: Which directions look most promising?

Hopefully, you now have at least one idea of how to make progress. Take some time to filter through them (if you have several) and decide which ones look most likely to work, or the least likely to give you a headache if you try them. Don’t spend too long putting them in order, just think about “this one is more likely to go somewhere than that one.”

6: What happens if you try one?

Pick your favourite idea and try it. What happens? Do you get stuck? Where? Why? Does it generate any more ideas? Does it generate any questions you need to answer before you can go any further?

7: Then what?

It might be that you succeed in getting to an answer you’re happy with. In which case, fistbump! It might be that you don’t. In which case, fistbump also – you had a shot at it. Nice work in either case.

But you’re not done - the end is not the end. Whether you succeeded or not, it’s still worth trying your other ideas: they might lead to a neater solution, they might lead to deeper insight, they might lead to you understanding something you didn’t understand before. This is the key thing for improving your problem-solving: looking for, and exploring, different ways to find solutions. The more you make it a habit, the better you will become at coming up with approaches, and the more solid your understanding of problems will become.

8: When should you give up?

You know, deep down, when you’ve put solid effort into solving a problem. (There’s a common theme in people asking questions on r/learnmath where they say “I’ve spent hours on this!” and it’s absolutely clear they’ve spent hours with the book open on the table while they watch YouTube.) If you’ve come up with many things to try and none of them have worked, it’s ok to put the problem aside. I grant you permission.

I don’t grant you permission to give up, though. Give yourself a break from it, sure. Give your brain a chance to relax and sort things out unconsciously. (Many of my best insights come just after I’ve started swimming, when it would be really inconvenient to write anything down.)

If you’re really stuck on a question, look in your textbook for similar examples. If you’re training yourself in problem-solving, I advise against looking up the answer straight away – struggling to find an answer is something you’ll likely need to get used to.

(That said, looking up the answer and then seeing how to get to it from different angles does have a place. Eventually.)

9: What did you learn? What can you do differently next time?

Probably the most important thing ((Yeah, I already said something else was the most important thing. Deal with it.)) is the post-problem-solving debrief. Hopefully, you’ve been working on a problem that wasn’t too obvious but also wasn’t too difficult for you. If you’re in that in-between sweet spot: fantastic – that’s where learning happens.

So what did you learn? Was there a trick you can make a note of to use again? Was there an insight that suddenly made sense? Was there something you forgot that you ought to remember in future?

Take some time to think about your problem-solving process and how you can improve it for next time.

10: Can you make the process your own?

This is my problem-solving process. You’re welcome to borrow it and use it as much as you like. You can even take it and make it your own. Think about where the process works for you and where it doesn’t. Are there steps missing? Are there steps you could skip?

Make a habit of practicing your problem-solving. Get into a routine so you always have an idea of where you could start. The more you do it, and the more you think about how you’re doing it, the better you’ll get.

Dear readers, if you have any other problem-solving suggestions, I’d love to hear them! mailto:colin@flyingcoloursmaths.co.uk and I’ll either add them to the post or steal them for myself. Ahem, I mean, save them for a follow-up post.

As for you, PROBLEM, I hope that helps!

- Uncle Colin