Question
How can I practice and further my studies in mathematics in my own free time?
Answer
I have found that self-study is VERY different from learning in a classroom. You cannot use the same materials or methods.
The best way to learn mathematics unfortunately, is in a classroom with a good teacher who is setting you weekly homework problem sets and challenging you with exams. Try and find a cheap course at a local community college. But drop out if the teacher is bad. A bad teacher is worse than no teacher.
The second best way is to get an informal private tutor (paid or a good-natured person willing to mentor you for free), who meets with you at least once a week.
The third best way is to find another person in the same situation and set up a weekly (or 2-3 times/week, the more frequent the better) self-study group, where you solve problems together.
If you don't put another person in the "learning loop" with yourself, it will be very hard. Nearly impossible. Things like Khan Academy can support, but not replace, a formal educational experience where others hold you accountable.
If you REALLY cannot find support like this, you have 2 basic self-study options that might work, from a psychology/motivation perspective.
If you were at a higher level, I would suggest picking unsolved math research problems to solve as your project, but unfortunately I can't think of any that you can attack without college-level math, unless you are a genius in the rough, like Good Will Hunting.
But seriously, don't turn to self-study unless you are absolutely forced to. Find a teacher, mentor or self-study partner. Your chances of success will increase 1000x even if a single good person is in the game with you.
For the record, I learned most of my math skills through intensive self-study in 11th and 12th grade, preparing for a very tough college entrance examination. But I did have the structured support from a school curriculum in parallel, even though I was teaching myself stuff that was way more advanced than things in the syllabus (I wasn't unique, everybody in my class preparing for that examination was about 1-2 years ahead of those who weren't). The school structure was not important for learning the actual subject, but was psychologically critical in providing the right learning support environment.
College and grad school for me were mostly class-driven, since I wasn't smart enough to self-study faster than my classroom work. For my PhD, I was forced to self-learn some advanced subjects and it was VERY tough, even when I tried auditing or sitting in on some classes. In the end, I learned the necessary PhD level skills using the "project" method above. Which means my knowledge of certain advanced subjects is very shaky and "project focused." I know some deep stuff that was relevant to the research I was doing, but I am probably missing some very basic stuff that I didn't need.
The best way to learn mathematics unfortunately, is in a classroom with a good teacher who is setting you weekly homework problem sets and challenging you with exams. Try and find a cheap course at a local community college. But drop out if the teacher is bad. A bad teacher is worse than no teacher.
The second best way is to get an informal private tutor (paid or a good-natured person willing to mentor you for free), who meets with you at least once a week.
The third best way is to find another person in the same situation and set up a weekly (or 2-3 times/week, the more frequent the better) self-study group, where you solve problems together.
If you don't put another person in the "learning loop" with yourself, it will be very hard. Nearly impossible. Things like Khan Academy can support, but not replace, a formal educational experience where others hold you accountable.
If you REALLY cannot find support like this, you have 2 basic self-study options that might work, from a psychology/motivation perspective.
- Get addicted: Pick a subject just within your reach (like 1 year beyond your current highest level), find a good book devoted almost exclusively to problems. Test-prep books are the best source. Subjects like trigonometry and coordinate geometry are good first addictions. Try the books of S. L. Loney, which contain hundreds of problems. Just keep solving them like you are playing a sport. I guarantee you will get addicted. The problems tend to be structured so skills build on top of other skills, so you'll develop a repertoire of tricks. Do NOT skip steps. Do not move to calculus until you can do trigonometric identities while drunk. Foundational math is NOT a subject like history. It is a sport to be practiced, like tennis. One way to structure this process is to sign up for some sort of certification examination a year or two out. That's what pushed me, and I would spend hours and hours alone, when I was 17/18, just cranking through one problem after the other.
- Find a project: If you don't have the stamina to develop an addiction, the other motivational scheme you can construct for yourself is a project. Pick something you really want to BUILD that necessarily involves math. Generally the best thing would be a programming project like a simple video game like Pong (pick a free, math-friendly programming language like Python). But if you dislike programming, make up some other math-heavy project like designing and building a model airplane from scratch. Then set short-term goals on a week-to-week basis for that project, and learn whatever math you need to get the next piece of the project done. I have to warn you: this approach will likely leave you with a foundational education that is unbalanced and full of serious holes (i.e. you will miss some basics that the project doesn't touch, and advance too far on other fronts without truly understanding what the hell you are doing).
If you were at a higher level, I would suggest picking unsolved math research problems to solve as your project, but unfortunately I can't think of any that you can attack without college-level math, unless you are a genius in the rough, like Good Will Hunting.
But seriously, don't turn to self-study unless you are absolutely forced to. Find a teacher, mentor or self-study partner. Your chances of success will increase 1000x even if a single good person is in the game with you.
For the record, I learned most of my math skills through intensive self-study in 11th and 12th grade, preparing for a very tough college entrance examination. But I did have the structured support from a school curriculum in parallel, even though I was teaching myself stuff that was way more advanced than things in the syllabus (I wasn't unique, everybody in my class preparing for that examination was about 1-2 years ahead of those who weren't). The school structure was not important for learning the actual subject, but was psychologically critical in providing the right learning support environment.
College and grad school for me were mostly class-driven, since I wasn't smart enough to self-study faster than my classroom work. For my PhD, I was forced to self-learn some advanced subjects and it was VERY tough, even when I tried auditing or sitting in on some classes. In the end, I learned the necessary PhD level skills using the "project" method above. Which means my knowledge of certain advanced subjects is very shaky and "project focused." I know some deep stuff that was relevant to the research I was doing, but I am probably missing some very basic stuff that I didn't need.