When I was doing math, let us say for example, introductory number theory, it seems to take me a lot of time to fully understand a theorem. By understanding, I mean, both intuitively and also rigorously (know how to prove or derive). However, I really find many results (even elementary results like euclidean algorithm or things like ax+by=gcd) hard to intuitively understand. I usually will be thinking about these theorems most of the spare time, whenever I am not doing anything requires my thought…
Even if I do comprehend, it takes a long time until I fully comprehend the theorem. And sometimes I may even forgot them(maybe I didn’t actually fully comprehend) I feel like I am spending tons of time more than others on number theory. I mean, many of people online claim that introductory number theory is easy. Is that I am not smart enough to do mathematics and make contribution to the world of math later on since great mathematicians must have great intuition, or is that other people are not fully comprehending the theorems and it really does take a lot of time to just figure out one theorem completely?
I am very confused. I like math, but I really want to know if I am capable of doing it and make contributions. And I wish to make the best choice for myself. I appreciate any good comments or advices!
Let me give you a personal story. As a young kid, I was always very strong in math but was pretty hampered by one of the worst educational environments in the USA. I ended up entering a magnet school for junior high and had to take a math placement exam to determine which of three math classes I would join: regular math, pre-algebra and algebra. I didn’t do quite well enough to fully justify being placed in algebra but did a bit too well to justify holding me back in pre-algebra. So as a seventh grader, I was placed in algebra. I struggled with it immensely. I had a private tutor and studied tons but to no avail. Ended up getting a 36 average or so and dropped down to pre-algebra in which I got a 105 average. Eighth grade came around and I had to take algebra; again, I didn’t do that great but was better than before. I ended up with a low 70.
I did very well in geometry in high school but again not so great in algebra 2. Pre-calculus was hit-and-miss: some topics I did very well in, some not so well in. It was not until calculus that I really began to understand math at an intuitive and deep level.
Since taking calculus, I’ve excelled in mathematics. I ended up with nearly a 4.0 GPA in my math courses in undergrad (one A-) and I am currently in graduate school doing pure math after all of the struggle I went through. I’m pursuing very difficult and unique research and am very fluent in various aspects of mathematics. Just because you are struggling now does not mean you are incapable. Plenty of good mathematicians had trouble with math at some point for one reason or another. Don’t throw in the towel so soon if you really like the material!