| > You can work through the concepts and delude yourself into thinking you understand something when really you're just hand waving it I keep seeing this and I don't quite understand how this one works. If I were studying math on my own (which I've done and still do), I'd do the following: 1. Pick a book. Say, Rudin's Principles of Mathematical Analysis[0]. Read a section, then attempt problems. Pick a problem. Say, "prove {1/n: n is natural} U{0} is compact directly from the definition(not using Heine-Borel)". It's guaranteed your "proof" is not a proof. 2. Compare your solution to existing solution manuals or ask a question on MSE[1]. Since the given book by Rudin is super-massively famous, each question has probably been asked/answered about a bagillion times each on MSE, so just searching MSE alone would likely to spit out many answers to your questions. People on MSE will tell you exactly why your "solution" is wrong and where you tripped up. Sometimes even the clarifying answers are hard to understand. But then you can ask new questions, think more, correct your misconceptions until it all finally clicks. Do that with all the rest of the problems[2]. I don't see how the process above is delusional. [0] This book is not a realistic fit for a novice, though. Instead, one would start with books like [3], [4], [5], [6] to learn how to prove things and think like a mathematician would. [1] https://math.stackexchange.com/ [2] In reality, the more math you see and do, the more mathematically mature and less dependent on others(to check your work) you become. In fact, if you can solve any problem in "B@by Rudin" and some famous abstract algebra textbook (say, Dummit & Foote's "Abstract Algebra") cold, you're way ahead of most any undergrad math major in the world! Because standards on undergrad math majors are not that high, nor that brutal the world over no matter what they say. If, additionally, you can solve any given problem in a book like, say, Hatcher's "Algebraic Topology" or any other famous grad level textbooks on, say, differential geometry or, uhh, functional analysis, you're officially in the big leagues. Again, if you're worried about being delusional about your proofs, you can always present them on MSE. [3] "Book of Proof" by Richard Hammach. It's online free. https://www.people.vcu.edu/~rhammack/BookOfProof/ [4] "Discrete Math" by Susanna Epp https://www.google.com/books/edition/Discrete_Mathematics_wi... [5] "How to Think About Analysis" by Lara Alcock https://www.google.com/books/edition/How_to_Think_about_Anal... [6] "Linear Algebra" by Kuldeep Singh https://www.google.com/books/edition/Linear_Algebra/BJNoAgAA... |
Because those that are just starting to learn math don't take your approach to learning.
I had a lot of bad habits that I had to break when I was learning math that really caused me to do poorly in many classes. If you are coming at it from a more qualitative field, then it's very easy to read the book and come away with nothing from it.
I'm taking a discrete math course and I was trying to offer up some advice on the school's subreddit to someone struggling in the class. They mentioned something about me taking the easier prof, so that's why I'm doing well. I said that I had fully worked at least 50 problems in each chapter we covered and asked how many they had done. The response was I went to lecture, read the notes once, and attempted the homework exercises; I didn't know you had to do some much work in this class.