TITLE: undergraduate math vs graduate math
QUESTION [42 upvotes]: It's really a mild, soft question.
So far, I am an undergrad student, contemplating on several majors.
What will be the major difference if I become a math major and go to a graduate school to study math?
Will a graduate math major generally focus on reading/studying recent research papers? Or will  the graduate math major be taught by professors on the things not covered in undergrad programs? 
The reason why I am asking this question is that undergrad math programs seem to me at this point so well-covered that after studying the programs, a person will be able to do anything he wants to do, and if one wants to do research, he will be able to catch up with recent progress by reading research papers.
If this is true, why is the graduate school even needed?
By the way, I am a first-year undergraduate student :)

REPLY [3 votes]: A short, but I hope informative, story from someone with a PhD in Physics, but who also was a professor of Mathematics for a few years.
I was teaching in the physics department at one of America's finest colleges and worked with a brilliant undergraduate math major and published three journal articles with him.  He got straight As, and seemed to be able to solve any problem I gave him... including one that I probably could have solved, but it would take me longer than he did.  He had such a bright future.
He was accepted to math graduate school at one of America's very top math departments but alas didn't thrive there.  There are a few fundamental difference between undergraduate education and graduate education, which he couldn't navigate.  (He left school to program computers for a bank.)  Speaking in broad terms:  

Undergraduate:  You learn math that is already understood by experts, and you solve problems posed by others (professors or books), and are under a fairly strict schedule of classes, exams, paper deadlines, etc.
Graduate:  You create new math that is fully understood by nobody else (yet), you must search and create your own problem, and have very few scheduling commitments, so you must be self-motivated.

This is not to dissuade anyone from attending math graduate school... quite the contrary.  I want to alert such prospective students so they know what to expect.  Actually, the best preparation at the undergraduate level is to do true research (appropriate to your skill and education) which involves creating your own math problem, and then doing your best to solve it.  
One of my criticisms of many undergraduate departments (not just in math or physics) is that they don't teach and encourage students to make their own problems.
So my advice to an undergraduate interested in math graduate school would be to meet with your math advisor and try to arrange a self study (or "independent topics" or whatever is the format at your school) and "test the waters."  You may love it (!) in which case graduate school sounds right for you.  If not, I think you'll have to decide whether you want to develop such skills (in preparation for graduate school) or instead pursue a career where others will give you problems to solve.  There is no shame whatsoever in such a career.  
A fun little diversion, which nevertheless illustrates the difference between undergraduate and graduate mathematics education.  Consider your favorite discipline in math (number theory, geometry, differential equations, ...) and just start asking questions/problems in that discipline.  don't worry that you cannot solve them!  (Fermat couldn't solve his Last Theorem, despite what he wrote in his margin.)  Just keep asking and asking them.  If you find you're stuck, or you can only come up with problems you've already seen, or if you're frustrated by such an effort, stop and think about that.  If instead you get energized, excited, and motivated by this little exercise--especially if you come up with a "gem" of a problem--then you may have the temperament and skills for graduate research.  (Of course this isn't the only requirement for success.) [You can search for my TEDx talk on "how to ask good questions" where I run through this exercise for sudoku puzzles.]
You want to follow a career path that gives you the greatest personal satisfaction and helps the discipline and society more broadly.
Good luck!