How should I learn maths?

Question

I am 17 now and I have several questions to you. To let you better answer my questions I think I should tell more about me. I am pretty good at maths at school(but I often make silly mistakes thats why I am not scoring only 5 <- that's the best mark in my country) I am quite good at school level but I really suck at competitions. My teacher makes me feel like I am not worth her attention and she is spending more time with students that are better than me thus making me feel like I am useless. Here are my questions:

1. Is math just really hard work and that's only thing I need to do?
2. I want to start solving exercises from school. Should I start from beggining of my book or from the point I am currently on in school? Or maybe I should solve problems from competitions( even if they are way to hard for me)
3. Should I consider making silly mistakes a big problem? How can I fight with it because I am really annoyed that I am getting a lot of 4s instead of 5s.

Answer 1

1 - The one things every great mathematicians have in common is their dedications to maths. Surely it takes some sort of genius to be able to be at the top level, but even the more gifted person at maths will never achieve anything if he/she doesn't put full effort into it.

2 - My advice would be to try and solve exercices of your level, and when I say solve, I mean perfectly solve. When I was younger I had intuition for maths, it appearred easy to me so I never really put the effort to be rigourous in my answer. While it might be ok if you're young, sooner or later you'll have to be dead on on your answer or you won't get full points. The sooner you start good habits, the better !

3 - First off, everyone does silly mistake. But there are ways to limit them to a minimum. One great simple technique I use is to recheck everything I write at the end of an exam. Hear me here, you don't only read what you just wrote, you redo the math aside and check if you get the same answer as before. If you find yourself lacking the time to do so, reading what you wrote can help you catch silly mistakes too.

Now for a couple of personal advices :

Maths is a vast, VAST, ocean. People spend their life studying it only to cover a small portion of it. You, got, time. Don't rush it.

Basics, basics, basics. There's a lot of cool stuff in maths and it's easy to find something cool like fractals and whatnot and try to understand it, but if you really want to learn something, you have to know everything that comes before it. If you read a definition and there is a word there that you don't know, just stop and go learn what that word mean, ignoring it is the recipe for disaster.

If you like competition, feel free to do it. But remember, maths is fun and ultimately it is about you understanding it, not how other perform.

It is better to get every single point for half the question, than half points for every questions.

Hope you have fun on your math trip mate =)

Answer 2

This became too long for a comment:

Have fun with it: Try to think up (relatively simple) problems that seems right on the border of what you think you are able to solve... You'll most likely need to learn some new techniques in order to solve the problem, and that's a great way of learning math! A famous mathematician, John Conway, very much has this approach to doing research (much of his most important work comes from him considering games).

The silly mistakes don't matter so much outside of competition math (and competition math doesn't really matter) - real math is about thinking abstractly and creatively; the algebra skills will come with practice, and your mistakes should become rarer.

And as has been said in the comments, don't worry about the scores. It seems like some of your motivation for doing math comes form a desire to be validated (which is perfectly natural, mind you!), but try to instead steer your mindset toward just enjoying math for the crazily beautiful and mysterious subject it is; enjoy the thrill of discovery, enjoy wondering about the deep structures of numbers, algebras, geometric objects, etc., and enjoy bettering yourself. Becoming good at something in life often gives a sense of confidence, which (partly, at least) makes the desire to receive validation and admiration from others less important.

Good luck!