One theory: Real Anaylsis.
I bought the textbook ‘ Introductory Real Analysis’, by Kolmogorov. I also find a course website maintained by instructors of the real analysis course, in Colorado, APPM 5450 and APPM 5440.
Nowadays, university professors post the syllabus, homework and solutions as well as part of the notes on course websites, especially for his student registered locally. This also unintentionally facilitates off-campus self-learning. If you have sufficient time and energy, just follow them and you can take a course just as a registered one in a college. That is also why the online course is so popular today. One even argue that this teaching and learning mode will dominate in the future, replacing the traditional classroom-based mode.
However, it is far less than perfect. There is no office hours. Perhaps you never know your homework is correct or not. In those cases, anyway, there is a way out. You can consult someone who has such expertise or seek solutions online. There are a lot of helpful people out there.
The main obstacle, in my opinion, is lack of motivation. At the very start, most of the learners would just keep a regular progress of the coursework. After one or two weeks, there may be one or two halts. Normally one month later, that learner, once so ambitious, may nearly forget what he is supposed to do. The motivation runs out, perhaps because no credit is there for accomplished homework, let alone the penalty for skipping them. When someone lies down and has a rest, and that does not incur any loss, why not take so much extra burden?
So, let me ask you a question, what is the credit for self-learning then?
Curiosity? Need for publish? Job seeking requirement? Or just pleasure? Beyond those urgent, short-term goals, there are some fundamental reasons, not merely for sake of material benefits. Such a reason lies in the heart of those people who did the theoretical work and wrote the textbook. It is a natural desire of seeking the unknown.
Desire is very different from only being ‘interested’. The well-known ‘Stay hungry, stay foolish’ is accepted by many people. Someone keeps such keen interest in biology, mechanics, computers until adulthood. But when they fail to continue the study, they may only feel guilty for a while , then get relieved. This interest-forget-relieve pattern is formed in this way. A habit that you fail an expectation of yourself, but never apologize to your heart. What can be worse than this?
To continue a self-study, you have to make sure you exclude all of obstacles that stand between you and the goal. At least, you should identify them. The motivation can be built on top of the daily habit. Do not make vows, just form a habit. That is all. You cannot fall asleep unless you brush teeth. That is a habit. You cannot sleep before you finish the problem set for your self-course. That is also a habit. It seems to be simple rule, isn’t it?
Yes, this blog essay is inspired last winter. But I failed myself again. For one more time, I stop the Lisp learning for 3 months. Now, what I can only do is correcting the wording and flows. Anyway, I hope this can help, at least for my academic writing skills. Also this tells that it is very difficult to keep learning without a regular course because a habit is not easily formed. Writing something down also does not work. That does nothing to form such a habit.