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leaf's the maths of morality course was the first supercurricular i ever did and it was really worth it! i think the best part of it is the discussion. i'd never really been around people who were 'into' maths so it was really inspiring to meet like-minded people. the staff and cohort also were so supportive and this is a really good way to find out about other supercurriculars as a lot of what i did later on was based on recommendations other people on the course gave me
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if you take physics, the COMPOS follow along programme is the next best thing i'd recommend. unfortunately, the main programme's deadline passed but honestly, i think the follow along programme would be just as valuable. the really good thing about this is that they actually use past STEP and PAT questions in assignments so this is preparing you for entrance assessments while improving your overall mathematical fluency
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if you're female or non-binary, it all adds up was a lovely day at oxford's maths department. this is just one example so i would recommend signing up to the mailing lists of the universities and colleges you're interested in, so that you can hear about taster days and sign up for one
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i believe that for maths, 'doing' stuff is slightly better than reading books (that's not to say don't read, but it's less important than for other courses. also, please don't put fermat's last theorem in your personal statement, unless you really genuinely adore it, as it's about the most cliche maths book you could choose) so doing any competitions is great - british physics olympiad and senior maths challenge are great
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it could be worth writing papers on any topics you're interested in and trying to publish them, even if it's only in a school newsletter. similarly, running clubs related to maths is a good way to show that you're interested in maths. realistically, you're not going to be doing anything ground-breaking at this stage; it's more important to show that you're able to learn and grow as a student, and that you have passion for your subject
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it's good to note that you don't have to only do maths-related things. if you can tie your other interests to maths, then that can offer a more interesting perspective that will help you to seem authentic in your personal statement. for example, it would be great to explore something like statistical biology, or to look into the relationship between maths and music
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when doing this things, try to have a critical mindset as this will help with writing your personal statement. i would recommend keeping notes of everything you participate in so that you don't forget by the time you get to writing your ps. it's important that your experiences build upon each other: ask questions and always try to explore things that you finding interesting more deeply

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