How can I (failing Cambridge's BA Math) improve my grade to a 1st from a 3rd-class honours?

Question

In my first-year exam for the BA Maths at the University of Cambridge in June 2017, I was the only one in my College to score a shameful 3rd Class Honours (in the past 20 years). My College warned that they'll permanently expel me on grounds of failure of academic progress, if I don’t improve. I:

1. hanker to score a First. But I don’t know how or what to improve! I'm becoming pessimistic and fatalist: the competition’s too crafty, and I appear too dumb, dim-witted. Perhaps I haven’t enough free will over my intelligence, and can’t smarten up!

2. love the probability and statistics papers of my course, but I like the abstract papers less. But this shouldn't be a snag, as after first year, I can pick courses to fit my interests.

3. attend all my lectures and supervisions.

4. study at least 8 hours daily. I feel that I’m already working at 110%! In order of priority, I read my textbooks and lecture notes, solve some of the questions in my example sheets, and then solve more problems from other books or past Cambridge exams.

5. eschew all non-academic, social events to focus on my studies. I sleep at least 8 hours daily.

6. spend at least 2 hours daily preparing and eating my vegan meals and fruits. My College has a dining hall, but the portions are always piddling. Even after eating there, I’m still hungry. I don’t eat there if the vegetarian option of the day is shoddy as it often is.

   I agree with my Directors of Studies, Supervisors that:

7. I answer the basics (i.e. first few parts of an exam question) perfectly. I remember my lecture notes and can do anything (e.g. theorems and proofs) previously seen. I don't have much trouble with computations.

8. But I can never prove anything novel (i.e. previously unseen), or calculate anything if it demands novel "tricks". Thus I didn't complete most second parts of questions. I'll spend a lot of time trying different ways to prove something or to apply previously seen theorems/techniques, but I never solve the question.

E.g., in my first-year exam, I was often to asked prove something not by induction. I spent 10 mins. trying one proof method (e.g. proof by contradiction), but failed to progress anywhere. Then I spent another 10 mins. trying a 2nd proof method (e.g. by contrapositive), but got stuck again. Then I spent the final 10 mins. trying a 3rd proof method (e.g. direct proof), got stuck again, and finally ditched the question to try something else, but never had time left to return to it.

Answer 1

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.

Courses

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.

Other Advice

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.

Answer 2

1. It's possible that you are at low end of the ability spectrum but given the thought in your post, I suspect that you can do better than 3rd.

2. I love probability and statistics (and plausable reasoning, inference, consulting, etc.) much more than abstract proofs also. Think Box, Hunter, Hunter. These are areas with huge practical application and in non research settings: biomedical, oil and gas, retail, etc. Try to vector through the BA thing you started (don't bail unless they boot you, almost ALWAYS better to finish things). But emphasize stuff you like. And then carve a path away from all the definition sharpness games and towards curious pattern finding.

3. OK, but try to emphasize being prepared before them, getting more use out of them. (see 4.5)

4. 8 hrs is plenty. Make sure you are productive, mind not wandering. Consider to take 5 minute break every half hour. Walk or just relax during the 5 minutes off. Don't do chores AND DON'T read or hit the net. You can eat or hit the can though. And don't compress all 8 hours into same time. Divide it out.

4.5. Try to read and work problems BEFORE your lectures. The lecture then becomes review.

Try to be efficient when reading the texts. Don't let mind wander--stay engaged (this can be mentally tough, but you can build up to it). Don't skim, but at the same time, don't lollygag. Also be engaged in the reading of the text but also don't get so super strict that you try to derive stuff out of the text. Work all example problems AS YOU GO, within the reading (not the hw problems at end of section but the atual examples in the reading.)

My guess is you are probably going too slow in the reading. Do it intense but fast. You are hungering to get to the problems (but at the same time, you will be taking them in quasi test practice mental frame, so making sure to understand the text...but move along.)

If the texts are ba$#-busters ("mature"), get some easier ones to use as side resources.

Write down questions from your reading (or hw) (use the back of your notebook, working backwards, leave space for answers). These questions should be asked and answered during your interactions with profs. (Some you will answer yourself as you read further or work problems). Write down the answer.

Try to work all the hw problems in the text, night before class. Check with a solution manual if available. Re-work any problems you mess up, from the start, treating it as an all new attempt. (Even if a dumb error.)

Schedule specific meeting with prof if you just totally get a block in an area (come with written specific questions, in notebook, and with written evidence of trying to work the matter out). After you get the help, don't just say "I got it now" but do more problems right after the session to prove it to yourself (get extra problem sources). If you still didn't get it after thinking you did, go right back.

Sounds like you are doing a fair amount of problems, but I still get a little bit the flavor of you doing a bit too much reading and not enough drill. Parsing lecture notes is generally not wise for instance. If the course is following the text, your lecture notes really should just be for unusual things: "watch out for this", "cool trick", etc. [Remember you have pre-read the text and worked the problems, before the lecture, so you are just reviewing in the lecture and learning nuances, answering questions.]

5. Think you are pushing too hard--don't usually say that, but you aren't doing well and are pushing very hard. Ergo! Add a Sabbath (non religious is fine) but a day off. Sleep is good.

Try to add 30-60 minutes exercise period per day. This can be light (doesn't need to be stress cardio...at max capacity) but some light walking or biking (or running, but not too harsh). Take it easy as you start. I recommend daylight and outdoors. Doesn't matter if it rains or snows, enjoy the outdoors. Get some vitamin Dshine. We are physical animals. Yes, we are.

6. 2hours per day on food prep and eating? And you have a food plan? Way too much time. Reminds me of a Navy XO who told us JOs when we complained about the POD, not leaving enough time for sleep: "if I had a choice between sleep and eat, it would be a no brainer". You can wolf the food during class breaks, bring it with you (doesn't mean it needs to be junk or get fat: finger salad veggies are very easy and filling). Use lunch hour for exercise.

If you are vegan for ethical reasons, stick with it but make sure you get enough whole protein, vitamins (take a multi), and essential fatty acid. If you are doing it for health or brain reasons, ditch that. Humans have been omnivores for millions of years. It will be much simpler and time efficient to get your nutrition and you run less danger of deficiencies.

7&8 (and general testing): This is tough. I would say, try to find some homework and do practice tests that (a) capture the feeling of novelty and (b) just expose you to a longer trick list.

In addition, the aspect of self testing (put a light pressure on yourself and think of it really as a practice test, not low stakes homework, drill or learning) can help with test anxiety. This is common in sports: "4th and goal, last play of the Super Bowl" yelled at the two sides of a scrimmage during practice.

Personally, I do this even for the homework at the end of a section. Not supreme self-pressure or frantic-ness, but a feeling that it is like a test or at least a sport or game. Not just slow drill. (this also helps time efficiency.) If you do it right, it creates enough pressure so the homework can be fun (game-like) rather than pure drudgery. And it gives a little bit of test situation prep. It still won't be total joy...but it helps. Note also (!) that the practice problems at the end of a reading section will have tricks or things that are NEW TO YOU at that time. (Even the simplest math texts have some problems that are trickier, the starred ones or the higher numbered ones. And for a Tripos smartie, I bet MANY of them are hard for you at first. So consider this as same situation you have during exams.)

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Good luck, man. You are doubtless a guy with a lot of talent so will have great jobs in life even if you flunk out of Tripos-land. But try to get through this thing and get the degree you are started on. It will definitely be something you are proud of in the end--the personal perseverance and guts, not the thing itself. Again, good luck!

Answer 3

I can hardly improve on the detailed answers here but just wanted to add a few things.

I have seen fellow students lock themselves way for the requisite eight hours of study and fail straight out of pre-med. Hours alone will not make it for you.

As in sports or physical activities it is not practices that makes perfect, but Perfect practice that makes perfect. More simply; hours perfecting the wrong grip for the wrong backhand will not yield the desired result.

Accessing other students, some more successful, some also struggling, will give you the latitude you need to spot weaknesses and problem areas you cannot yet report.

I cannot speak to vegan anything except to recommend you get a great deal of protein by any and all means. You will pass nothing without it.

In working on proofs. Knowing intellectually each of the approaches does not volunteer the answer. It is seeing the endpoint clearly that will show the path. This sounds impractical as a process to follow but it's the truth. To get such a high level view you need more exposure to other people and the way they think about things, including math. Not even the best biography of mathematicians will let you share their insights. But a drinking buddy, a cooking group, a PhD bartender (I had one) will let your mind develop in ways that mastery of the text never will.