# How do undergraduates do mathematics?

> Source: https://ollybritton.com/notes/books/how-do-undergraduates-do-mathematics/ · Updated: 2024-10-04 · Tags: book, notes, uni, uni-book

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> A guide to studying mathematics at Oxford.

This book is actually free, you can find it [here](https://www.maths.ox.ac.uk/system/files/attachments/study_public_0.pdf).

- 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 [redacted](https://ollybritton.com/404) 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?“

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