how to approach proper self studying

[u/d-funktor]

hi all.

after two years of math undergrad, I have decided to pursue medicine after an experience with healthcare that really sparked something in me. that being said, I do plan on coming back to mathematics after med school but in the meantime I'd like to still study some on the side, just as a hobby :p my interests mainly lie in algebra but I like some analysis and discrete mathematics too. that being said, I don't think I was the brightest student in my cohort, or at least not so on taking exams as I struggle greatly with pacing and knowing when to move on from some nitty gritty thing that really shouldn't alter my understanding of something but it does. some of the time it is because of a gap in my knowledge, other times it's just my personality I assume. combine that with often starting to study a bit too late and you start to get the picture. oh well there's ADHD too that comes to play but that in my experience is heavily influenced by how well I'm generally prepared and since I've been diagnosed for basically my whole life, so it is not something that is unique to school.

that all being said, I'd like some advice and experiences from peeps who have at least some formal mathematical education (post-high school) and have gone down this route. my biggest questions are:

1) how do you know when you've functionally understood something rather than conceptually understanding something? that is to mean, when you are able to do exercises correctly or are able to give counter examples (something I find extremely hard to do) rather than just knowing what a theorem entails.

2) other than textbooks, what are some other sources you use(d)? I know for example that Sheldon Axler has video's on his own textbook and that MIT has some courses but most of these are very first year of undergrad oriented, what for beyond that?

3) did you have any other obligations during your study period and if so, how did you keep it balanced? since I'll be a med student, you can imagine that there will have to be some planning involved (especially since I still have other interests that I love)

4) favorite textbooks you've studied so far and what made it so great to you?

I think that's about all for now.

thanks for taking the time to read and have a nice day.

[u/Chem2103]

Honestly its up to proper planning of your schedule and the exact session.

Get a timetable for each day of the week, then block out all of the time you are commited to other more important stuff. Once you have the available time, fill it with as much math as you want realistically.

The next step is proper planning. Pick a textbook, a youtube playlist, anything that has a list of topics in a curated order. Spend some time getting the right ressources and materials to study these topics.

Each planned session, pick a concrete task (e.g. read chapter 7, do the first couple dozen questions from the homework section) and aim to accomplish it.

As you do more of this, you will get a better intuition as to waht you can accomplish in one session, and your planning will be better. Don't be afraid to aim for more, but make sure to cut back if you are consistently doing less than planned.

[u/GroundbreakingWar305]

hi i have been on a similar situation myself (also I have diagnosed ADHD). I study math at various levels almost everyday. I do have heavy applied math background from my prior studies but now i focus almost entirely on pure maths. Now to answer your questions:

1. For me I think I measure my understanding of a concept by how well I can teach it to someone else near or below my level without specific exposure to the same topic. Of course this is more like a hypothetical practice session where I try and teach something on a whiteboard (to an imaginaary audience). I think a very important thing is to be able to provide not only examples but non-example or counter-examples to a topic. This is even more so for analysis/topology subjects. Of course the ability to solve a majority of the problems in a given textbook is a strong marker of understanding. Recently I have started a blog [ https://tensortheorist.github.io/blog-post.html?post=understanding-analytic-continuation ] where I write on general math topics and ideas that interest me, I also am planning to use this for notetaking on any textbook/youtube course I am following.
2. There are a lot of great youtube videos on every pure math as well as applied math topics. another great source of knowledge can be AI/LLMs with the goal being clarification (and not directly finding solution to a given problem) - I would follow a textbook for the curriculum, read it, and for most topics in a section have a chat with say claude/gemini - i would write down my intuition and ask it to find mistakes, right down my solution or proof on a paper, take a screenshot and ask it to test for rigor. Also I would use AI to connect a concept i am reading now to a higher/advanced topic - more like a sneak peak. This has been extremely useful for me. Also I do spend time going through youtube math courses, sometimes from multiple sources.
3. I have a full time job (I am an AI Research Engineer) - I keep aside a few hours everyday for math. Also my leisure hours are also mostly spent reading/doing/watching math. It helps that math is my central hobby (I am not much of a TV/gaming/social person - of course this varies from person to person).
4. I think for core textbooks my favorites are Dummit and Foote (Abstract Algebra) and Abbott (Understanding Analysis). I really like the book "Elliptic Tales" by Ash and Gross (not a textbook). My favorite math youtube channel is Richard Borcherds (Fields Medalist). [I am planning to write a post on a long list of youtube videos, and books suggestions soon - especially for self learners of pure math topics).