Hey [name] (probably not a good idea to put your name up, btw). I'm not entirely sure this is an appropriate post for the forum, but I think it could be helpful.

Anyways: we've all been in those shoes before. STEP seeming like a impossible mountain to scale, A-Level maths seeming too easy; typical Cambridge mathmo story, so you've got as much of a shot at doing well in STEP and getting into Cambridge as anyone else.

(i) Stop depending on your teacher(s). It's 2016, you've got loads of resources available and A-Level maths is.... well, it is what it is. (not allowed to use the word trivial on here) - this is good practice. Most of your maths learning and understanding will come from sitting down and banging your head over something till you understand and/or get used to it, not from a teacher/lecturer teaching you everything and you not having to do anything. In any case, most schools cover material way too slowly, most candidates will end up needing to teach themselves C4 or M2/S2 or whatever early in order to be able to start with STEP Prep.

(ii) Grab textbooks and/or online lectures - when I was self-teaching myself A-Level maths, I thought examsolutions.net was pretty good for developing the basis of my understanding. The EdExcel textbooks are a little meh, from what I remember, but they suit some people (I assume you're on Edexcel, if not substitute the name for whatever your board is), attempt easy STEP/MAT questions (more so the MAT at this stage - you really need to know C3/4 before doing STEP), use this forum to ask for hints on problems that you attempt. Try out fun problem sites like brilliant.org that really help your problem solving skills, attempt SMC/BMO1/whatever relevant competition maths is for your level (I'm not a fan of this, but you might be the sort of person to enjoy it). That said, take it easy - you've got plenty of time, you'll be surprised how much you can improve in 6 months or a year with some hard work.

(iii) STEP isn't as big of a hurdle as it seems. You're currently in lower sixth and STEP does seem impossible: trust me, I know. But it's not, it'll get a lot easier the more you do it; come back and look at a paper after you've had some more experience with A-Level maths and you've matured a little mathematically. STEP is the very last hurdle to getting into Cambridge and I'm not sure it's entirely wise to be thinking about it this early!

Hi there!
I'll second most of what Zacken said, with a few caveats: first, I personally didn't find A-level maths as easy as Zacken seems to suggest it was. Maybe you'll find it easy, maybe you won't, but neither is a strong indicator of how suited you are for Cambridge. While, of course, they like students who are good at math, that's tested largely at the interview and in the STEP papers rather than through A-level results, so if your sixth form doesn't leave you with the best A-level results, Cambridge will almost certainly still consider you - especially if you make them aware of the issues with teaching.
Secondly, I found the Edexcel and AQA textbooks really useful. They cover all the required material, of course, and make a reasonably large range of problems immediately available. Certainly I wouldn't recommend them as the only resource to use, but if you can ask your college to pick up some copies for you to borrow - or if they already have some - I certainly found them very useful.
Sorry to be persnickety, Zacken!

All that being said, I'll agree that it's not as impossible as it might first appear. Cambridge are very good at accounting for individual circumstances (as long as you tell them, of course), and I believe there's even a question in the pre-interview questionnaire - whatever it's called - that asks if your textbooks and teaching were good enough, so rest assured that this won't disadvantage you.
I'll try to give my own advice; mostly repeats of what Zacken's already said, I guess, but maybe it'll be useful to you.

(i) Speak to your teacher. Try to communicate that you aren't engaging with their teaching style, and see if you can work with them to find solutions to that problem. Teachers want you to learn, so see if they can help you do that. In particular, if you want them to expand on something, or to clarify a point, it is very helpful to ask questions in lesson, regardless of whatever instincts that may go against

(ii) Speak to the Cambridge Admissions Office, or whichever college you're interested in applying to. They know what they want from you, and what you'll have to do to achieve that, so they can probably give you some strong advice.

(iii) Definitely, definitely learn using outside resources. Textbooks, the internet, whatever. If you ask around, it's possible that other local colleges will have support programs for further maths and STEP students, so if you ask your college to interface, you can probably get onto one of those. These forums should be a good resource as well. However, there's only so far other people can get you. Try to do as many hard problems independently as you can, and only look for hints here and elsewhere if you get really stuck; trying to find the solution on your own gives you insight into how to come up with chains of logic in the future, which can help break down blocks in your thinking

(iv) Personally - and this may not be a problem for you, in which case ignore all of the following - one of my biggest issues with self study was that I would understand some concepts or topics fairly quickly, assume I knew how to do them, and then not try questions or even revise the topic thoroughly. Even with practice exams, I'd see that I could "see the answer", and not bother working through step by step, which would miss some crucial points. I've since become better with this, and have two main techniques for it: first, I take thorough notes (from the text), and if something's not clear, expand on it myself. This ensures you've actually processed the information, and understand every step. Second, even if a question seems "easy", or a step seems "obvious", answer it fully. If there are any ambiguities in a question, you want to be sure you can resolve them before you're sat in an exam room. On a similar vein, if you don't immediately see a solution, particularly in STEP, it can be very easy to look it up, get to grips with someone else's solution, and assume that understanding the solution means you can solve the question. That is very much not true; unless you answer the questions yourself, you can't be sure that you'll get them in the exam.

(v) Read around the subject! There are plenty of introductory texts on, for example, formal proofs, basic number theory, or elementary group theory, and exploring interesting new maths can help keep you motivated when you're slogging through piles and piles of integration questions. Brilliant.org and the like are also good here.

(vi) Relax a bit - while STEP is a difficult paper, it's intended for the end of your A-level career, so worrying about the impossibility of questions now isn't necessarily the best of plans. And, as Zacken says, it gets easier with practice. Aside from getting good (enough) grades in AS-levels, there isn't much you need to do before next year, so enjoy yourself!

(vii) Remember that Cambridge is not the be-all, end-all. The UK - and abroad - have plenty of universities able to give an excellent mathematical education, so if you slip up on STEP, it's not the end of the world.

Of course, take this with a pinch of salt - we're mathematicians, not life advice counsellors, so any advice we give is suspect. That said, best of luck with whatever the future holds!
-BHB

I can relate to your experience of maths class at high school. At my school we were often taught "recipes" to apply to answer questions with little explanation of what was really going on. For example, we were taught the power rules for differentiation and integration without any justification and how to do matrix multiplication without any explanation of why it is defined the way it is. This way of teaching may be effective in giving students the necessary knowledge to pass GCSE and A-level exams, but it is poor preparation for STEP and university-level mathematics, as well as leaving students with a deficient understanding of the subject and obscuring much of its beauty.

As you may be aware, edX (at edX.org) provides lots of free online courses from some of the best universities in the world in a range of disciplines. MIT provides three courses, 'Calculus 1A: Differentiation', 'Calculus 1B: Integration' and 'Calculus 1C: Coordinate Systems & Infinite Series', that are intended to teach calculus from scratch: nothing more than knowledge of high-school algebra is required. To teach yourself calculus you could do much worse than try these. The concepts are carefully explained and many of the results, including the power rule, are proved.

It's clear you have the most important requirement of all, the will to learn, so hopefully, we'll see you in Cambridge!

There is a series of YouTube videos on the channel 3Blue1Brown intended as an introduction to calculus. Watching these should be a good way to improve your understanding of the ideas underlying the calculations you do in school.

https://youtu.be/WUvTyaaNkzM?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr