How to become a good mathematician .
Hello Aspirant,
Mathematics is a field whereby to become a professional, i.e. a mathematician, one must really get a Ph.D. As such, my advice would be to take as many electives in mathematics as you can, e.g. real analysis, topology, abstract algebra, and then, when you graduate, apply for a M.S., or if you are very advanced (you can check this by trying qualifying exams online and seeing how well you do) a Ph.D., in mathematics of course. It might be worth switching majors before graduating, but you could also just do your undergraduate degree in Telecommunications, if you'd like.
To avoid a rocky start in graduate school, one should at least have a thorough knowledge (at an undergraduate level, of course) of all the GRE topics: calculus (single and multivariable calculus including vector calculus, differential equations, and of course pre-caclulus), linear algebra, abstract algebra, and some select topics (I recommend topology, real and complex analysis, discrete mathematics, and probability with elementary statistics, but this is somewhat overkill).
One should also be comfortable with more advanced topics in a field or fields you think you might want to do research in. If you really loved your abstract algebra studies, study some representation theory or commutative algebra, for instance.
Before you begin going down this road, I suggest you look into what it is mathematicians do, though. You might be surprised to see what research is like. Doing an REU or similar program would be a great idea.
