Ok so I suppose if your College was at all typical then after your exams they suggested to you that you might want to transfer to a different university/subject. And as you are still doing maths at Cambridge you told them you would like to continue and presumably you have them some kind of indication that you would be changing your ways. As lent term is about to start I’m guessing that you are not making sufficient progress.
First up the bad news (although perhaps you already know this): Transfers are much easier after first year than after second year. If you fail this year (or according to your question get a 3rd) then you won’t be in Cambridge next year. It’s also likely harder to transfer to another university. It’s basically too late to switch to natsci or economics so if you want to stay, you’ll have to do it through maths.
Now some advice:
First, lent term will start soon and in second year that means CATAM. It’s a good opportunity to pick up marks and those marks can push you over from a fail to 3rd or a 3rd to a 2.ii. Doing CATAM May also help refresh your memory about courses you’ve already done. On the other hand, much of CATAM requires one to have the ability to prove some things so that may put you at a disadvantage but you should still be able to pick up a decent amount of marks. If you haven’t started yet, please, please do start now (and don’t leave the Easter term one to the last minute—you want to be revising at the start of term, not doing CATAM).
Next up, some advice for maths: You say that your problem is in proving things. Unfortunately almost all questions involve some proof. Pure questions definitely involve proof. Stats questions likely involve proofs and applied questions usually involve some element of slightly informal proof. The first year Numbers and Sets, Groups and Analysis (and Met and Top if you count that) are meant to be your introduction into pure maths/doing proofs. I would suggest you go back to proofs/exercises from lectures/example sheets from Analysis (and maybe also Numbers and Sets, and Groups). By studying these perhaps you can get a better idea as to how to do proofs. In Analysis a key idea to get into your head is that almost every proof actually only has one idea and every other step is “you look at what you have and there is only one reasonable thing you can write down, so write it down.” The hard part is building the intuition as to what “reasonable” means.
When you write a proof you need to avoid the mindset of “I need to figure out what formula to apply and then plug things in.” That works for A-Levels but it almost never works at Cambridge. Therefore don’t have a checklist “First I’ll try proving by contradiction, then if that doesn’t work I’ll try proving by contrapositive, otherwise I’ll try induction, and so on” because you’ll spend all your time flailing around not knowing what to do. What one should do is this
- work out what it is your trying to prove (or maybe disprove)
- think about the statement and conditions
- maybe a counterexample exists. Think about what a counterexample might look like. Maybe you find one or maybe you think of something close and learn something about the problem by doing it. I think this is an important point. There’s no point trying to prove something that’s clearly false
- think if any theorem you know might be relevant to the statement (in Cambridge maths exams that ask you to prove something, this is almost always the case but it isn’t necessarily obvious which theorem to apply or how or both. You build intuition about this by doing related problems). If you think of a relevant theorems, think what you might need to apply them. Think about what you would need to prove to be able to apply the theorem. Maybe you can use the proof of the theorem that you know as a skeleton for the proof you want to write. Maybe thinking about this gives you a plan to prove the thing you want to prove.
- if you still don’t know what to do, try negating the statement, (ie proving the contrapositive/by contradiction), or just look at your assumptions, look at what you want to prove and write down everything you know. Now you have some other things to look at so apply the above steps again.
It may be worth getting hold of a copy of How to Solve It as that might help you with proving things.
In second year the only stats/probability courses are Statistics and Markov Chains. You might have some luck with stats. If you can’t learn to prove anything (and especially if you never really learnt v&m last year) then you’ll probably struggle with markov chains. You probably know that anyway. In exams you might be able to pick up a few algorithmic questions in markov chains but it’s a short course so there won’t be too many questions.
My recommendation is that you should definitely do Methods and Complex Methods. If you can learn those well and understand them, you should be able to pick up a decent amount of alphas in exams—certainly enough to pass. They don’t rely much on proofs and if you can learn the algorithms alone you should get some alphas from it.
You might be able to get something from linear algebra if you can learn the algorithms but you need to have an idea about how to do proofs to write answers to these questions (excepting some betas maybe but it isn’t worth learning a huge course for a few betas)
If you have the right mindset you might be able to get some marks in fluids but don’t be afraid of dropping the course if it’s not going well.
I don’t know if you did VP/Optimisation last year. Both of those are worth a shot (although perhaps only if you feel up to doing a course in exam term). Optimisation is just learning algorithms and doesn’t need so much proof.
Quantum mechanics is variable. You might get lucky and get some questions that just need calculus but you need to learn the methods, have some intuition about (quantum) physics but be able to turn it off, and sometimes you’re just flailing around with weird questions about operators and commutators which a prelude to PQM in part II.
It’s too late for you to take up GRM now.
Talk to your DoS. They will have good advice on which courses to take and how to strategise to gain marks in exams. Don’t focus on stats/Markov chains if they aren’t going to get you marks in tripos. You should be able to pick up some marks from doing lots of short questions—they just need you to know the definitions and maybe a few main theorems. You don’t get many marks from trying and failing to do a long question. Read the ones for courses you’ve studied, you should have a rough plan for how each step of the question will go and how hard it will be. Start with the easiest question and continue until you finish it or get completely stuck (in which case move on and maybe come back). It is inadvisable to begin a question if you haven’t a clue as to how it will end.
Talk to your tutor. They may be able to help you and give you advice as to what things you might do next. If you do poorly in exams it’s much better for you if you had appeared to be looking for help/struggling. If your tutor suggests intermitting you should seriously consider this (SFE will pay for up to four years towards a single degree at a single university. The degree doesn’t have to be specified which allows one to switch subjects at a single university and for ones “undergraduate” BA to be merged into a MMath to do part III without extra funding. I would strongly suggest that you do not rule out intermitting for reason of wanting to be able to get a loan for doing part III)
Maybe it’s worth talking to someone in your JCR about this although I can’t think who this would be or how they might help.
Believe it or not, maths is a social discipline (even at Cambridge) so I’d also suggest talking to the other people in your college about maths (or not). This can help internalise the knowledge, turning over definitions and theorems in your head and getting a feel for how they feel. This may be as little as talking g to people on the walk back from lectures (if you walk) or going to hall and talking to people over lunch (if they go to hall after lectures), even if you don’t want to eat the food there. You may have heard about some high-performing mathmos who seem to lock themselves into their rooms and come out with top marks in tripos but do not be fooled: most of these people talk amongst themselves and thus may improve their intuition and knowledge from the ideas of others.
Make sure you have a good diet. Being vegan makes this difficult. You won’t be able to think properly if you aren’t getting proper nutrition. Ideally you shouldn’t need two hours prep time for food either. If you really do then maybe there are some other vegan students also being disadvantaged by this prep time and you could pool you’re resources.
Make sure you get a chance to relax/socialise. I’m not suggesting you should go out and join the boat club (or indeed any society) but if you spend all your time stressed about your poor performance then you won’t be in a good state to learn. Perhaps there is some society relevant to your interests. Many are relaxed and provide light activities and light socialisation and don’t care if you don’t turn up all the time. You could also go to society talks from the Archimedeans or TMS or scisoc. If I remember rightly the Archimedeans will have their traditional pizza/board games night in the CMS after the CATAM deadline. It’s worth being on the mailing lists even if you don’t think you’ll be interested.
In the CMS on Saturdays there’s a “maths cafe” which you can go to. Just turn up with some work and there’ll be a long table with people and some water and stuff sat around. You can do your work here and there’ll be people who you can talk to about your problems and what you are stuck on. They should be able to help. Even if (think) you don’t want to talk to people I suggest you just turn up anyway, just in case.
You should get notes from your supervision (or your partner). Hopefully you can use them with what you remember from the supervision to be able to get the questions you didn’t have previously. If it’s too hard to remember what was said then you might want to ask your supervisor if you can record the supervision (use your phone as a voice recorder). You probably won’t learn much from merely going to a supervision.