First you need to learn about mathematics. Read general books and surveys; browse a good encyclopedia of math; get a feel for how the different topics fit together. Two good general books are Davis and Hersh’s The Mathematical Experience and Eric T Bell’s Mathematics: Queen and Servant of Science (this has the problem that it is dated and doesn’t even mention computers and has little on probability and statistics, but is still worth reading for algebra and analysis). Also check a college catalog for required courses, and read the detailed description of the courses, what they cover and how.
Second, unless you just want to learn about math, you need to practice. Mathematics is a skill, and you must practice it to be able to use it. Work examples, work problems, create your own problems to illustrate questions you have.
Third, for self-education, get multiple textbooks. If you run into problems understanding a specific technique with your primary text, read what other books have to say. Different authors approach problems from different angles, maybe a different approach will help you. Old textbooks can be really cheap, there is no good reason not to have several different versions for subjects that are important to you. This is less important if you have someone you can ask about problems, but can still be important, not many people are very good at explaining what they know.
[edited—I had misremembered the authors of The Mathematical Experience]
Read general books and surveys; browse a good encyclopedia of math; get a feel
for how the different topics fit together.
I disagree, but not very strongly.
If you find yourself interested in a topic, focus on that topic. Your interest and motivation is more important than being well-rounded or comprehensive.
Most people who are interested in math for itself already know enough that they aren’t going to be too interested in this sort of post. For the majority who learn math for other ends, engineering, computer science, physics, etc, browsing will help them orient themselves, and to find out what interrelationships there are between different fields and techniques. And what they are missing when they discover they need background that they don’t have for a book they are working through.
First you need to learn about mathematics. Read general books and surveys; browse a good encyclopedia of math; get a feel for how the different topics fit together. Two good general books are Davis and Hersh’s The Mathematical Experience and Eric T Bell’s Mathematics: Queen and Servant of Science (this has the problem that it is dated and doesn’t even mention computers and has little on probability and statistics, but is still worth reading for algebra and analysis). Also check a college catalog for required courses, and read the detailed description of the courses, what they cover and how.
Second, unless you just want to learn about math, you need to practice. Mathematics is a skill, and you must practice it to be able to use it. Work examples, work problems, create your own problems to illustrate questions you have.
Third, for self-education, get multiple textbooks. If you run into problems understanding a specific technique with your primary text, read what other books have to say. Different authors approach problems from different angles, maybe a different approach will help you. Old textbooks can be really cheap, there is no good reason not to have several different versions for subjects that are important to you. This is less important if you have someone you can ask about problems, but can still be important, not many people are very good at explaining what they know.
[edited—I had misremembered the authors of The Mathematical Experience]
I disagree, but not very strongly.
If you find yourself interested in a topic, focus on that topic. Your interest and motivation is more important than being well-rounded or comprehensive.
Most people who are interested in math for itself already know enough that they aren’t going to be too interested in this sort of post. For the majority who learn math for other ends, engineering, computer science, physics, etc, browsing will help them orient themselves, and to find out what interrelationships there are between different fields and techniques. And what they are missing when they discover they need background that they don’t have for a book they are working through.