How do I tell if I am able to go to graduate school in math? This is my first question on this site, and this question may sound disturbing. My apologies, but I truly need some advice on this.
I am a sophomore math major at a fairly good math department (top 20 in the U.S.), and after taking some upper-level math courses (second courses in abstract algebra and real analysis, differential geometry, etc), I can say that I genuinely like math, and if I have A BIT chance to succeed, I will go to graduate school and choose math research as my career.
However, this is exactly the thing that I am afraid of. My grades on the courses are mediocre (my GPA for math courses is around 3.7), and for the courses I got A's, I had to work very hard, much harder than others to get the same result, and I often get confused in many of the classes, while the others understand the material quickly and could answer professor's questions, and at the same time I didn't even understand what the professor was really asking. I really wonder, if I have to work hard even on undergraduate courses, does that mean I am not naturally smart enough for more advanced math, especially compared to everyone else in my class? Can I even survive graduate level math if I even sometimes struggle with undergraduate courses? I always believe that adequate mathematicians could do well in their undergraduate courses easily. In my case, even if I work very hard, I forget definitions/theorems easily and then of course forget how to use them to solve problems.
Is it still worth to try if I am significantly behind the regular level and have to work hard even for undergraduate courses, providing that there are a lot of smart people who can understand them instantly. This feeling hurts me a lot, especially when I am struggling with something in math, I always feel I am a useless trash and ask myself why I am so stupid?
I thought about talking to my professors about this issue, but I find this too embarrassing to start. I am really afraid that if I ask them this question, they may tell me the truth in person that "you are really not smart enough to go to graduate school".
So how can I tell if it is still worth for me to think about this path, or I should realize that I have no chance to succeed and give up now? I appreciate encouraging comments, but please, please be honest on this case because it is really important for my future plan. Thanks again for your advice, and I am really grateful.
 A: The answers here are rather idealistic. They seem to be based more on trite cliches rather than concrete reasoning or evidence. 
The fact is, academia is competitive and jobs are scarce. Your grades matter.  Your performance relative to your peers matters. Being passionate at math or being interested in the subject is a necessary but insufficient condition to succeed in graduate school. 
I'm going to be brutally honest with you: do yourself a favour and don't pursue graduate school. I'm sorry if that's your dream, but we must have a sense of realism. It's nothing more than several years of daunting work and, in return, you get to call yourself a "mathematician". Job prospects? Salary? Unexceptional, if you're lucky. 
Learn to program, and get a job in that field; people who are strong at math are almost invariably good at programming. If you're struggling and genuinely think you're incapable, I'm not going to lie to you and say that passion and perseverance is necessarily going to fix everything; it might, of course, but that's a conclusion you must reach, perhaps with the help of your professors. 
I'm not saying any of this because I think you lack the talent; I'm saying it because it's the advice I'd give to any friend, unless he or she is genuinely a genius. Indeed, I'm making this same decision myself. 
This answer may well be down-voted, and that's fine; it's simply a consequence of this site's demographics. Responses to this sort of question will invariably be opinionated, but, nevertheless, it is an important one. 
A: Hopefully I can answer before the tide of "talk to someone who knows you personally" and "this question is off topic" rolls in and your question is inevitably closed. It is true that you should talk to someone who knows you better, but I can give you some general advice that is better than just "follow your heart."
First, you should know that professors see students like yourself, who have a low opinion of their own ability, all of the time. We get questions like "am I cut out for this or that?" quite often. Your situation is not as unique as you might think, which, I hope, is comforting to you.
Here are some issues to consider.
Not all graduate schools are the same, and not all career paths are the same. As a Ph.D. in mathematics, you might become: A career lecturer at a large public university, a research mathematician, a professor at a small liberal arts college, a teacher in the math dept. of a private high school, a simulation/modeling expert in thermodynamics for a large oil conglomerate, a control systems analyst for Boeing, a data analyst for Facebook, a quantitative analyst at a large wall street firm, a mathematical ecologist for the united states government, etc. etc. etc. All of these areas are well served by mathematics Ph.D.s but require hugely different skill sets that you develop while you are in graduate school, and after you leave graduate school.
There are many different graduate schools. Some you may be well prepared for, others you may not be. At some well-known graduate schools, we have people take real analysis and other upper level undergraduate courses during their first year of study. There is no shame in this! The point is whether you love what you are doing and whether or not you are making consistent, quantifiable progress. If you are working really hard and not getting anywhere, that is bad and you should consider a career shift. If you are working hard and making progress and love what you are doing, then for goodness sake keep doing it!
Here is how you should approach your professors. Ask direct questions that have answers. Here are some examples: Based on my performance in your class, do you feel that I would be a successful first year student in graduate school 3 years from now, supposing that I make the same level of progress that I've been doing? If so, do you have recommendations for specific schools? Your professor might hear this question and go off on a tangent, or become alerted to something else in your tone, etc... and give you more or less advice than you wanted. But the fact that you asked a specific question is much better than "Am I cut out for grad school?", which is a question that essentially has no answer.
In sum, your education is not wasted. Keep learning. You will thank yourself in 10 years.
A: Often what happens in mathematics, from my personal experience, is that the one with more exposure generally gets the upper hand. What you are talking about here is talent, which you feel you lack. Talent plays a role till a certain extent, but then I have seen talented people too struggling with high level mathematics (algebraic geometry, to be precise).
So here is what you should do. You have realized that your performance in class is a bit mediocre as compared to others, which is a good thing. Now, it is time to better your prospects as a math student.


*

*Read in advance before the class. You do not know how important this is. Try to get a hang of what is going to be taught. Get a textbook, read the things before you come to class. Understand what is going on. Go through the examples. Solve exercises. This is probably the most fundamental important thing students miss.

*After doing 1., when you come to class, you will get a better understanding of what your professor is going to teach. You too will be able to answer his questions, mark my words. If you cannot think of an answer immediately, write the problem down, think over it and at the end of the day, if you are unsuccessful, go to your professor. Tell him what you tried, and why they failed. Don't just go with a problem and say "I couldn't do anything with this."

*Thing is, the very good mathematicians I have seen, are extremely well-read. Read a lot. Spend time with maths. Understand how topology is related to geometry. These interconnections between math topics is very crucial.

*Discuss. Your discussions may happen with your own professor, or with your fellow students. Do you know, I never really understood what is the use of Measure Theory. Why the hell would someone make easy things so dry? And then I started talking to my professor. I told him to teach me. I believe the long evening hours the two of us discussed in his office have been crucial to my understanding of mathematics.

*I would advise you to do the above things for a sufficiently long time, before you decide to switch to a career other than math. Believe me, math is fun. You love it, and you are already in the minority. All you have to do now is sharpen your skills.

*Research thinking is something you should develop. You can read some accessible papers on advice of your professor. The method to read an easy paper is to try to prove the lemmas yourself, after probably skimming through the main ideas. Don't read word by word. Can you give an alternate proof to some lemma? Can that topic be generalized? Talk to your professor, again!

*Think, read and study! Do this for the next few months and then comment on your progress again.

A: At some point, math has less to do with intelligence and more to do with patience and methodology in learning. 
If you really want to continue studying math, then why not "try" grad school. For me, grad school was less about getting a master's degree and more about learning more mathematics.
Overall, grad school was one of the best experiences in my life thus far. I was immersed in mathematics, and surrounded by like minded folk. Math comrades make for the most enriching of friendships. 
If you are in awe of mathematics, then, at the very least, why not go to grad school to behold more beauty? 
