This question is about getting realistic expectations for how a university student actually does and should learn maths. I'm becoming increasingly suspicious that my approach is detrimental, but I don't see a way out of it. I know that screams "off-topic personal question", but I think the themes of this question should appeal to a broad audience and can be answered without personal knowledge of who I am and how I learn. Please don't automatically close this; I will endeavour to edit this into a more answerable shape if necessary.
Tl;Dr - skipping the context, my question is more or less what it says in the title. I learn things very carefully but there is a massive time overhead involved. I would like to find some sort of compromise position, but don't know how to change. I could restate the question as: "how to take better notes?" since my current note-taking strategy is rather intense.
Mathematics is supposed to be a rigorous subject. As long as you don't delve into foundational issues - as long as you can believe in a bare minimum of basic set theoretic constructions and basic logic - you can, in principle, understand and prove beyond all doubt any statement you learn in your journey. This is what draws me to mathematics beyond any other subject; anything I read, I am empowered to question and (attempt to) check on my own. Despite my relative ignorance, I can point out mistakes in the author's work and decide for myself what the correct statements should be and I can know with certainty (ignoring foundational issues) that it is right.
When I started getting into higher maths (i.e. beyond school level) I was aware that "in 'real' maths, we prove things" so I made an effort to learn how to read and write proofs; every new thing I learnt I would either try to prove for myself or look up a proof that I could understand. There is considerable additional effort spent in expanding upon and annotating proofs with further explanations - Lord knows authors leave a lot of missing gaps for the readers to fill. I kept this attitude up; it has only been in exceptionally rare cases that I have noted down a statement without noting its proof. I have only done so when the statement was a side remark invoking a theorem that was way above my paygrade at the time. Even then, I would have tried (and failed) to understand the relevant papers.
As you can imagine, this is a very, very slow process. I've somehow managed to cover a lot nonetheless, but it can sometimes take me hours or even weeks to get notes on a single paragraph - there's something I don't understand, so I spend time trying to check it, then I find I maybe need to learn some new things before moving on, etc. and before you know it, what feels like an eon has passed.
- On the one hand, this is a good way to expose myself to new maths and new ideas, and by thoroughly checking everything written and filling in the gaps I get experience in "research", overcoming frustration and doing my own proof writing, problem solving. Moreover, by writing down annotated and fully explained proofs I have a reliable reference so that whenever I need to revisit a theorem and its proof, I can have more confidence that (a) the result is correct, because my younger self checked it and (b) I don't need to waste extra energy checking mysterious steps in the proof, since my younger self has hopefully explained them
- On the other hand, this is absurdly slow (I check and take the time to write down absolutely everything) and everyone I've talked to on (mathematical) social media seems to not at all follow the same procedure. I get the impression most maths students - very successful ones - 'get away' without painstakingly examining and cross-examining their textbooks. It seems many people don't even read the proofs of many of the statements they might learn in a textbook, preferring to mostly focus on the ideas, the intuition; this seems to work wonderfully for them.
- On the third hand, authors are capable of mistakes and omissions. It is not common but it is not uncommon either that I encounter sloppily stated theorems/proofs or false/grossly misleading statements. One benefit of my approach is that I can usually detect this and correct it; had I just had pure faith in my learning source, I may have learnt a false thing and missed out on the opportunity to learn/discover new mathematics necessary to make the corrections. This phenomenon gets worse as the maths gets harder and more abstract, I've found - which motivates to be more fastidious, essentially making me learn slower and slower
The 'fear' of learning false statements and blindly accepting mistakes from authors is with me, and it's why I struggle to 'grow out' of this approach of mine. However when I read things like this I really get the sense that I've been doing it all wrong, very wrong. I start university this year and I should probably start on the right foot; I don't want to be left behind in the dust by less fastidious - but faster - students.
- Is this slow approach genuinely problematic, or am I just being dramatic?
- How can one get through content more quickly but still manage to learn proof techniques and feel like they understand everything that's going on? I fear that if I started, say, skipping the proof of every other theorem, I would quickly lose track of how things are working and I would be afraid of failing to catch mistakes/sloppy statements
Perhaps the question becomes more answerable if I state it like this:
- In your experiences of teaching and learning, what are some good strategies for learning mathematics at a reasonable pace, all the while maintaining a high (ideally perfect?) standard of rigour without getting bogged down?
In case this context is needed; I’m not following any course (not at university yet). I’m entirely self studying; at the moment this has been investigations into enriched and model category theory and simplicial homotopy theory, following a few different textbooks.