How can an autodidact gauge their learning progress when relating to mathematics?

Question

Over the past few months I have been learning about discrete mathematics and proof writing. I understand that learning writing proofs is a collaborative effort - I need someone to actually review my proofs and check and see if I am doing things correctly. However, I don't really know where to find someone to do that for me so my approach here has been just gathering a few resources (different books) that have answer sets that I can check my proofs against.

Besides proofs, I go through different resources and do exercises (also checking my answers against an answer key), but it is difficult to gauge as an individual how much progress I am making. The only thing I can think of is to gather a bunch of exams somewhere on the internet with answer keys and to take them if I were taking them "for real" in a classroom somewhere. Is there any other way?

(A little bit more context for my case in particular, I am looking to improve my math knowledge to have a better understanding of computer science, for which I am also self taught)

Answer 1

This is a direct benefit of being a student at a university; you have more experienced mathematicians available to check your work and to see if you've not grasped a given concept correctly.

In some sense, formal courses restrain students as well since most students are inclined to follow the material within the course and don't seek to learn things outside of it unless circumstances force them to. There's already relatively little time during the semester to deal with the class material, let alone deal with other viewpoints on the same material.

I am currently a student in a university but before that, I was self-learning a lot of math. It's not easy to test yourself but I tried to do some things that helped me.

There are two suggestions I have regarding this.

1. Try to do some 'hard' proofs and post them here to get them criticized by people. Some math textbooks have a designation for problems that are especially hard (3 stars or whatever) so try working through those and post your solutions here. You'll get evaluated on what you know or don't know. It will give you a sense of what you can improve in your mathematical writing and it will also give you a sense of what you understand/don't understand.

When I was self-learning Analysis, for example, I would often post my suggested proofs for most of the big theorems of Analysis in One Variable. I would ask for feedback regarding those proofs and that helped me quite a bit. You don't have to do this but it's great to do it.

2. Often, there are multiple viewpoints on a given topic in math. Like I mentioned above, students often have to explore these in their own time because course time doesn't really give them the space for that (if you got the space for it, then great!). You, however, are learning this stuff on your own and have the luxury of 'time'. So, what you should do is learn other viewpoints while also working through your main text.

For example, say that you want to learn some Topology. Many Topology books focus on defining topologies via open sets (you don't need to know what this is) and that is just one approach to dealing with continuity. However, there is another approach that deals with continuity via nets and this isn't often covered in introductory topology books (maybe these books might include exercises on nets and so on but they certainly don't play a big role in the theory that is developed).

So, learn about 'continuity' using nets, for example. Prove some of the theorems. Get some practice with viewing the same subject using a viewpoint that you're not currently studying as your main viewpoint. This is always, always helpful because perspective is a good thing to have in math (it may give you new insights or better ways of viewing something). It also tests you out by forcing you to pull out and confront material that you think you've understood, giving you space to reconcile it with the alternative viewpoint that you're learning about.

I hope the example above wasn't too far-fetched. I am, after all, self-studying topology so I'm only speaking from my limited experience with one or two mainstream topology texts :) In any event, I hope this advice was helpful to you and good luck with your studies :D