First I will give some background, to put my question in context.
I am currently in my final year of school, and in order to meet my university offer to study Mathematics this Autumn, I am required to sit the STEP Mathematics Exams this June. For those who don't know (which I assume will be the vast majority of people who read this), there are three STEP (stands for "Sixth Term Examination Papers") exams entitled STEP I, STEP II and STEP III respectively, each 3 hours long. STEP I,II share a common syllabus, STEP III's syllabus is broader. See here for a detailled specification. STEP II,III are intended to be harder than STEP I. They are graded U,3,2,1,S (S being the highest). There are 13 questions, each worth 20 marks. You may pick any question you wish to answer and the best 6 solutions count towards the final mark (out of 120). Examples of STEP exams would be: STEP I 2001, STEP II 2002, STEP III 2009.
These examinations are used by the best UK universities (Cambridge, Warwick, Imperial, etc) for mathematics to provide a better discriminator between the ability of candidates. The reason they choose to use this exam is that A-Level Mathematics exams are more computational and do not require much mathematical thinking. In contrast, STEP exams are more proof-based, or at least closer to "real" undergraduate mathematics.
My aims for this exam
I am aiming to be able to write 5-6 full solutions in the 3 hours, but currently I am struggling to adapt to this more formal style of mathematics (which I appreciate is less formal than university mathematics, but the style of questions in STEP is considerably more formal than A-Level).
My questions:
- Is there anything else I can do besides practising questions that will improve my ability?
- Are there any books that you think will be particularly helpful?
- Is there any advice that you think will be useful?
Thank you for taking the time to read this.
I was recommended to work through the book "How to think like a mathematician" before starting my undergrad. It focuses on the transition from high school math to university math, in particular how to write proofs. It might be helpful for you to go though it. Otherwise than that, just practise, practise, practise.
EDIT: One more thing: if there are other people you know taking the STEP papers, have a few sessions with them where you talk about your solutions and share ideas. It helps to work through the problems once on your own, and then once in a group.