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
• 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?“