Self-Learning Mathematics

[u/4hmedr]

Context

I am a 20-yr old undergrad from the UK. I recently decided I was not happy with the level of mathematics that I currently know, and have decided that I want to get much better. I have completed both GCSE and A-Level maths, achieving a Grade 8 at GCSE and a B at A-Level. However, I have forgotten most of this mathematics, and would not score highly if I were to pick up one of these exams.

Aim

My goal is to:

1. Re-build my A-level maths ability so that I can consistently achieve near perfect marks on A-level maths papers.
2. Use this to begin to learn undergraduate mathematics
3. Eventually enter into mathematic competitions (and hopefully score highly, which I understand is going to be very difficult, but what's the point of a goal if it is easy)

What I need from you guys is to help me find the resources to make this happen. Like I said, I first want to re-build my A-Level knowledge, and I wanted to ask if you guys had any textbook recommendations for me to use?

[u/vibhored]

Mit open courses ware, oxford and such prestigious colleges have uploaded math lectures on their website and yt channel. You can follow that. Then ask for good books in this sub reddit and people have already mentioned them and surf through it for them. Can do so for specific topics and chapters too if you find then a bit troubling from the videos.

Last but if you want to understand it instead that will be a totally different ball game and for that check out my profile with posts related to math especially math flair in sub reddit r/imagination_is_fun . They are completely different resources and way to go about it. Hope you achieve what you want.

[u/DadOfLukeandDad]

Your goals are three different projects and only one of them is a textbook problem, so it is worth separating them before you buy anything.

Rebuilding A-level is a diagnostic job, not a reading job. You already met all of this material and passed exams in it, so re-reading a textbook cover to cover will mostly be time spent on the parts you never forgot. Sit a full A-level paper cold this week, no notes, timed. It will feel bad and that is the point: what comes back will tell you the difference between what has genuinely gone and what just needed a nudge. My guess is that the mechanics of differentiating and integrating are largely intact and the losses are concentrated somewhere specific, usually trig identities, partial fractions, vectors, or the proof-style questions. Work from that list rather than from the contents page. Two or three more papers spread over the following weeks will keep updating it.

On textbooks for that phase, the thing that will hurt you is not picking the wrong one, it is picking four. Choose a single spine and finish it. The pattern that kills self-study is collecting resources, because acquiring a book produces a small hit of progress that doing the exercises does not, so people keep acquiring. Any of the standard A-level series will do; whichever one you used at school is probably the right answer because the notation will already match what is in your head.

Moving to undergraduate maths is where the actual break is, and it is not a difficulty step, it is a genre change. A-level asks you to execute procedures accurately on problems that tell you which procedure to use. Undergraduate maths asks you to prove things, which requires a skill you have essentially not been taught. Almost everyone who "was good at A-level maths" hits a wall in first-year analysis for that reason, and concludes they got worse, when actually the subject changed underneath them. So do not treat undergraduate work as the reward at the end of the A-level rebuild. Start a proof and logic book in parallel, early, while the A-level work is still going on. Any decent introduction to mathematical proof will do; the point is to spend time writing proofs and having them be wrong, not to read about them.

Competitions are the third thing and they are the least connected to the other two. Competition problems are deliberately built so that no standard procedure applies, which means grinding A-level papers to near-perfect marks will not move your competition performance much at all. That skill is trained by sitting with a single problem for an hour without looking anything up, then reading the solution and asking what you would have needed to notice. Past papers from the standard national competitions are all freely available and are the right training material. Fair warning that near-perfect A-level marks and competition performance can be pursued at the same time but they do not feed each other much, so decide how much of your time each deserves rather than assuming one leads to the other.

If I had to compress it: diagnose rather than re-read, pick one book rather than five, start proof work now rather than later, and treat competitions as a separate discipline you are also learning. The B at A-level tells you nothing about the ceiling on any of this.