# How do undergraduates do mathematics?

A guide to studying mathematics at Oxford.

This book is actually free, you can find it here.

- Pattern of work
- Lectures
- Most of the time, make an outline during a lecture and then spend time going over the notes at the end
- If a lecture gets hard to follow, try and write down everything that the lecturer does, and anything else that seems important at the time. This makes it easier
- Hard to follow lectures are often more useful because you absorb the content more the long run since you need to work through it
- After lectures, attempt problems in the problem sheets. Most problem sheets will cover the content of multiple lectures but it’s useful regardless
- Don’t spend large amounts of time stuck on one problem, come back to it later
- If you read ahead in the recommended reading, it makes the lecture easier to follow, and attending a lecture makes reading the book easier
- Try to resolve any obscure points, e.g. by talking to other students

- Tutorials
- The tutor is there to help you
- Tell the tutor what you have struggled with because it’s the quickest way
- Take away anything that the tutor writes down
- Meet up with your tutorial partner before tutorials and work out what you have both struggled with in order to save time
- Afterwards, you should rewrite complete solutions to the problem sheets if you got them wrong and fill in any gaps in sketchy notes
- Verbal communication with tutorial partners is often better because it forces you both to explain yourselves

- Private study
- Vacations are not holidays
- Should complete any problem sheets leftover in the first or second week of term
- Go over old content in the last term
- Read ahead for the next term’s courses
- Collections after a vacation are your tutors way of making sure that this gets done

- Total working time should be that of a full-time job
- Should get into a regular pattern of work as early as possible
- Find a regular place to work, free from distractions
- Find other students with whom you can discuss your work

- Lectures
- University mathematics
- Compare lecture notes with textbooks to try and get multiple different viewpoints
- Make sure you understand the formal statement
- Compare different versions of the theorem; does each imply the other? is one version more general?
- Try to prove the theorem yourself without reading any given proof.
- Make sure that you understand the given proof line-by-line, i.e. you understand the meaning of each statement, and why it follows from the previous statements.
- Identify where in the proof each assumption is used.
- Identify the crucial ideas in the proof (mark them in the margin of your lecture notes, or note them on a seperate sheet; this will be useful for revision).
- Try omitting one of the assumptions; does the conclusion of the theorem still hold? can you find an example to show that it does not?
- Try the statement of the theorem and the proof on some special cases to get a feeling for what it means.

- Any proof is mostly just fine details surrounding one or two crucial ideas in the proof
- Try and remember the essential ideas rather than the whole thing
- Do problems frequently, e.g. between lectures and especially while reading
- A long list of things to try inspired by
[[How to solve it]]
^{?}by G. Polya. - A useful test for if your proof is well structured is to try and read it out loud, you should be able to.
- Ask yourself ”if my solution were printed as a worked example in a text-book, would I find it helpful and easy to follow?“

- Compare lecture notes with textbooks to try and get multiple different viewpoints