How do I tell if I am able to go to graduate school in math?
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.
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.
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?