I think a lot of you are over-thinking this. Fortunately for maths, the personal statement doesn't matter a great deal. In fact, no tutor is going to appreciate a beautifully crafted statement more so than one which is less eloquent, so I wouldn't worry about it massively. It certainly didn't when I was applying for courses (although I didn't know that at the time), and at least Imperial confirmed that when I emailed the admissions tutor a couple of years ago.
Incidentally, his advice is that its more useful to have one or two concrete examples of things you have done to understand what maths is about, than to list a load of books you've claimed to read but not really got all the way through. Those examples may be:
- Reading books (my advice: only talk about a couple, and include a brief sentence or two explaining what you got out of reading the book). If you're struggling for choices, my favourite is "What is Mathematics?" by Robbins and Courant, but it's a little heavy going compared to most. URL="http://maths.cam.ac.uk/undergrad/admissions/readinglist.pdf"]Here is a comprehensive booklist.
- If you had to self-teach further maths, definitely highlight that
- Participate in competitions (this most likely means the Senior Maths Challenge). Don't worry if you haven't done well (the Imperial tutor was very optimistic in saying that "he'd be impressed if someone got to the last stages of their national competition... as if such a person wouldn't opt for Cambridge over Imperial!). If you have been part of the mentoring scheme, or did well in more junior competitions, that's good too.
- If you are planning on taking MAT, AEA or STEP, mention that too. Beware though that some universities (such as Imperial or Bath) may give you a higher offer including STEP if they know you're taking it and are not sure whether to make you an offer
Additional examples, which I expect are very rare include the following. Don't freak out if you don't have any of these examples. I doubt very many people can say the following
- If you were selected for any "Masterclasses" (I went to some from the Royal Institute, don't know if they still exist).
- If you have attended any additional courses from a university, or even if you went to talks from a university lecturer. If you live near a university, you might want to get your school to email them to see if you can attend a course if you're dead keen. Courses start in September / October and will likely clash with college so this is not going to be an option for most of you. See however if they have talks or the occasional lecture that you can attend.
Put in all the stuff about career aspirations and why you're interested in mathematics (I don't see anything wrong with the word "passion" but maybe listen to Slowbro as he's probably more clued up than me). But honestly, the examples is the thing they'll note and move on. It's hard to convey genuine enthusiasm in an arbitrary setting like a personal statement, so don't worry about it too much. When we had a results party for graduation, my Imperial tutors read out examples of personal statements and all were shocking. A Cambridge tutor whilst drunk took the mick out of peoples personal statements. They know these things are crap; I wouldn't waste too much time trying to get it perfect.
I would however make sure that the spelling and grammar are perfect though, there's no excuse for getting that wrong in the statement.