I should emphasise that the PS is far from a priority for maths. That borne in mind, the structure I used was along the lines of
- Intro: a deepish-sounding notion of what mathematics 'is', or what core feature you love about it. Best if it's concise, original and memorable. It may be appropriate to link this in to a (perhaps non-technical?) maths book of your choice, though preferably not a clichéd one (where clichéd here means 'written by one of Simon Singh, Marcus du Sautoy or Ian Stewart'
.
- I followed this up with some stuff about the maths Olympiad stuff I had done up to that point, plus a little about my formal qualifications. (Note that while lying is normally a bad idea on personal statements, elision and ambiguity can work wonders if correctly applied.) A teacher once told me 'Don't state - relate', in the sense that while merely saying you got a Gold in the SMC or whatever is a bit dull, following this up with '... which I could not have achieved without my natural ambition' or 'which inspired me to learn more about combinatorics' or whatever is a bit more useful for whoever is reading it.
- I then talked a little about specific things I had looked at beyond the syllabus. Were I to rewrite mine I would be a lot more specific than I was; rather than saying 'I extended my knowledge of integration' it may be more fruitful to say that 'I looked at DUTIS/reduction formulae/dimensional analysis methods*' and going into a little more detail. The risk is that you'll be asked to show that you can walk the walk as well as talk the talk in the interview, so don't mention it unless you're rather confident.
- I then used a chunky paragraph to discuss my experiences with physics, which made some sense in my circumstances I assure you. If this applies, it may be appropriate to couch this more as applied maths than as physics (the two, after all, are asymptotically equal).
- I also discussed my maths mentoring experience. In this sort of scenario it is important to ensure that you discuss the benefits that proceeded for you, rather than the students in question - for example, concise explanation, communication and so on.
- My school insisted that I had a paragraph utterly unrelated to maths, which was probably a good idea in fairness. Just throw anything in here that makes you look interesting. Perhaps it was not a good idea to do as I did and try to link activities to mathematics in a cringe-inducing manner: I find it doubtful that anyone would be convinced by my assertion that my DofE expeditions improved my mathematical ability through the experience of practical problem solving, but who knows. It may yet succeed.
- A short concluding paragraph is generally a nice idea; essentially restate in a couple of sentences why you're a good applicant, convey your passion and round it off by summarising what you hope to get out of university.
That ended up longer than I anticipated - rather like my personal statement, in fact, so be prepared to judiciously trim. In the end, it's unlikely to be a major factor in the decision, though it may well be used in interviews as either introductory general chatter or as a lead-in to some maths you may have mentioned.
Feel free to let me know if you have any more specific questions.
*sadly underrated
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