Anatoly Vorobey (avva) wrote,
Anatoly Vorobey

самостоятельное изучение математики (англ.)

На форуме кто-то попросил совета: как изучить самому математику в рамках университетской программы (первой степени, бакалавра), если нет возможности собственно учиться в университете? У спрашивающего есть знание математики в рамках средней школы.

Разные люди дали ему разные советы; вот мой (по-английски, переводить неохота):

  1. This isn't easy. Look for something to steady your motivation. Question your reluctance to go through the formal process (i.e. go to a university), weigh it against the substantially higher risk of "dropping out" of self-directed study.

  2. If you live near a decent university, look for ways to use it without attending it formally. In many places, you can audit the courses freely without anyone caring that you're not a paying student. Use the library, not just for the books, but to force yourself off the internet and to study, if you have a problem with that. Knock on the door of a math professor during their office hours and ask them to help you prepare a self-study program (don't trust any of their recommendations blindly). Or do the same with a PhD student. Don't think that you're intruding. Mathematicians as a rule feel that anyone without a college-level background in math is missing out terribly in life. They will nearly always be sympathetic to someone trying to gain that knowledge.

  3. Try different ways of acquiring knowledge; look for the one that clicks with you. Some people are very strong at studying from a textbook. Others learn much better from a lecture, and there're many excellent university courses available as video online. If you're studying from a textbook, be sure to try and do the exercises. There are some very few people who can learn without doing exercises; you're almost certainly not one of them. Most mathematicians, including most brilliant mathematicians, aren't either.

  4. Treat any textbook recommendation, here or elsewhere, as a weak hint. For whatever reason, even though the contents of undergraduate math textbooks on a given subject are almost completely determined by tradition (and trivial from a professional point of view), different textbooks seem to resonate strongly with different people. Even the most famous texts don't work for everyone. If you hate the text several people told you is the best, it may be wrong for you; try a different text (w/o compromising on content).

  5. Go to the website of a top university. Study the degree requirements for a math major. They will have a number of required courses; find their course pages, look through the syllabi, note the textbooks the instructors have chosen. Compare with 2-3 other top universities. Use this data to design your own course of study based either on detailed lecture notes/videos of the actual courses, or on the textbooks they are based on, in the latter case make sure to cover everything in the course's syllabus.

    Example: start with Examine the lower division required courses in detail: For any given course, google its name and "berkeley" to find an even more comprehensive course page for a particular version in a particular year, e.g. It might have the complete list of textbook chapters to study, exercises to do, homework/midterm examples, etc. Make use of those. You now have enough information to structure the study of this particular course for yourself, if you can muster the necessary mental discipline.

  6. You cannot skip (basic) things you don't like. For example, you can't skip calculus, and just go on to study other things you like. You won't have the flexibility of abstract thinking that comes from running the epsilon-delta formalism a few hundred times in your head, in different variations, and you'll run into a mental wall later on in any other subdiscipline of math. Basically all the material in the required courses list in the example above is something that is a) blindlinly obvious to any professional mathematician; b) should be, if not blindingly obvious, thoroughly familiar to you by the time you're done with your course of study. All of it is real core stuff, something that either lies directly in the foundation of most advanced math material, or helps you develop thought patterns necessary for most advanced math material.
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