I wrote this when I was excessively bored during exam term of my first year. It may grow as I get better at working (I’m something of a revisionist). The advice is entirely Cambridge-based; a lot of it probably applies to other places with minor alterations. Most of this comes from personal experience.

During a supervision, your supervisor will be writing all the time. As soon as you leave the supervision, mark the sheets that are particularly important in some obvious way (eg. by colouring in the corner). That way, when you’re frantically flicking through the notes at the end of the year, you’ll see where the information you need is. By “most important”, I mean the places where the supervisor explains something fundamental to many questions, rather than the ins and outs of one particular question.

Use Anki during the course - after each lecture, add the key factoids to your Anki deck for that course. (It’s a bit annoying to do at the time, but it’s seriously so much easier this way when it comes to the end of the year.) Try and get into the habit of doing some Anki every day. Remember that Anki does LaTeX!

If there’s anything you don’t understand, email your supervisor quickly. Some of the supervisors are absolutely brilliant at replying to emails; but all of them will reply eventually. If you have a DoS [director of studies] who is all-powerful (my first-year DoS was head of almost everything in the maths department), most of your requests can be granted (even possibly to the extent of shuffling lecture times around at the start of the year, if given plenty of warning and a very good reason).

It’s going to feel weird at first, but you almost certainly aren’t the best in the year - you’re likely to be average. (That’s what “average” is usually taken to mean - pun not originally intended.) This means that the lecturer probably isn’t interested in hearing your pedantry or requests for rigour during the lecture. It’s the supervisor’s job to clear up points that you didn’t understand. If everyone you’ve spoken to doesn’t understand something from the lecture, then it might indeed be the lecturer’s fault; in that case, email the lecturer, or go down to speak to them at the end of the lecture. If you notice something wrong that the lecturer’s written, then unless you’re absolutely sure the lecturer’s made a mistake, check with the person next to you before calling it out. Protocol is to wait for a brief pause in speech before shouting “Should this bit be this instead of what’s on the board?” - try and be as specific as you can, saying (for instance) “In your statement of Theorem 16, the first line says “f is differentiable” - should that be “g is differentiable”?”. Most people are not specific when they spot a problem, and it makes it much harder for the lecturer to diagnose the problem if they don’t know exactly where the problem is.

Don’t let your sleep cycle get too out-of-sync. It’s absolutely fine (after the first couple of weeks of the term, anyway) to go to bed at whatever time you’re tired - in my experience, everyone else is also tired and welcomes the chance to sleep. This is put on hold during the first couple of weeks of the term, because that’s when everyone’s all excited to be there and there’s not too much work.

If you have anything impairing your work that your DoS could conceivably help with, raise it as soon as you can. The earlier your DoS knows about it, the earlier something can be done, and your DOS is paid to worry about this sort of thing.

If both you and a friend are having trouble working, go together to the library and work next to each other. You might find it helpful to view it as a competition between you and them, or as a “suffering in comradeship” kind of thing. Maintain an absolute rule of “no talking to each other”, though. Schedule a break every 45mins or so, go outside and stretch your legs, and at the start of each 45min block you can ask each other about things you got stuck with on the previous 45min block.

You will not be able to do every question on the example sheets [problem sheets you do as homework] easily. You’re expected to have a good go at them all, but not to complete them (that would be a bonus). For those questions you can’t do, pretend you are in an interview: write down your thought processes, what you’ve tried and why it failed. Pretend you’re trying to appear really intelligent and solution-seeking in front of a prospective employer.

In your answers, use lots of words; your answer should not just be a list of equations, but a coherent argument. It’s a hundred times easier to mark if you explain every step properly, and it means you can go back over it at revision time; it’s not that hard to do at the time, too. If you find that you pick up your work before a supervision and have no idea what you’re wittering on about, you need to make your answers clearer.