20

Memorization per se should not be the primary focus. When I learn something new, I type up notes on the computer in a reverse-indented outline format. Then as time goes on and my understanding improves, I edit the notes to reflect that. When I work an exercise or read a paper, I refer back to my notes. Memorization, to the extent that it happens, is just a ...


17

One of the comments above mentions "the huge increase in cost" for using color in a book. The large cost increase for using color in a book was true twenty years ago. However, now the cost differential is quite modest. My book Measure, Integration & Real Analysis, which was mentioned in the question as an example of a math textbook that uses ...


16

For context, I have a lot of experience self-learning mathematics. I spent a summer learning additional algebra, point-set topology, linear algebra, and analysis (to extend my undergraduate degree) before entering my current graduate program. This was sufficient to skip a literal year in the program. I am now well into the program and have to self-teach ...


4

I teach at community college. That means I have a masters degree. When I first taught statistics, I had never taken a statistics course. Luckily, it was before I adopted my son. I spent 60 hour weeks, studying from many different textbooks, and asking myself questions. (I proved some results that we would never prove in our stats class.) I was an ok teacher ...


4

For most people, including myself, there is pretty much no chance that you’ll remember most of the things you’ll see and do in math-land the first time ‘round. It’s just an inevitable relic of the fact that mathematics is... well... hard, and it isn’t something that the human brain was designed to do. The fact that you’ve gotten as far as you have is an ...


3

You don't describe engaging in conceptual chunking. You mention the Sylow theorems as an example and I will also use them as an example. It is easy enough to memorize the theorem statements. If you do not chunk the material in the proofs, then I challenge any claim that you understand the proofs. (Fix a positive prime integer, $p$.) The Sylow theorems ...


3

There is no generally applicable answer to your question. In my own career I often taught while learning the material as my mathematics department began to offer more and more computer science. I learned the material a course at a time by signing up to teach it. I talked a lot with colleague mentors who knew the material. I always told my students in the ...


2

I think it is important to do some drill EVEN (maybe especially) with advanced concepts. This is because the concepts may be more strange and abstract. So you need to do some basic work to get familiarity with it. Quantum mechanics (or E&M) are rather non-intuitive and you just need to work with them. Not everything is as much an "aha" ...


2

One way to remember the mathematics that you learn is to create a narrative that explains how the concepts are threaded together. This narrative need not have any historical relevance (and it usually cannot, since historical developments are usually tangled and messy). The goal is to lay down a road from start to finish, passing through all the important ...


1

I believe you should follow these steps: Understand what the theorem says, with some applications of it (which also means to do the exercises, also show your work to TAs) Try to prove it yourself and get stuck quickly Work through the proof and try to understand it Take a fresh sheet of paper and now try again to do the proof When you get stuck (which you ...


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