Am I just not smart enough? [closed]

Solution 1:

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!

Solution 2:

When you say that you're experiencing some difficulties understanding intuitively some elementary things, there are a couple of possibilities:

1 - By "understanding intuitively", you actually mean the point in which you have devoted enough time and effort to certain topic that everything becomes clear and straightforward. That is what understanding something really means. Most people don't reach this stage as they stop their learning process when something "makes sense".

2 - If you regard "intuition" as an immediate understanding in the same way we know that 1+1=2, let me tell you that most mathematical concepts are not amenable to that kind of intuition. As timur said, many concepts in mathematics don't have parallel in the real world. Therefore, you can not expect the euclidean algorithm and 1+1=2 to produce the same "result" in your brain. As Von Neumann said:

Young man, in mathematics you don't understand things. You just get used to them.

That's what happens. You get used to dealing with very elaborate concepts and they become second nature to you.

Finally, if you forget something you spent a great deal of time studying it but after refreshing your knowledge, you are able to regain that understanding quickly and somewhat effortlessly, it means you actually understood it very well the first time. That's how the human brain works when it comes to nonessential things.

Solution 3:

When I learned number theory, I found that I had no intuition for anything about the proofs, where my classmates seemed to pull the things from thin air; in math, there's always that little (or not-so-little) brilliant leap required for the proof. When I just started learning number theory, I had no idea how people were figuring these things out - even knowing the proofs, I found them hard to follow. I don't really think I got up to speed with my intuition until months after initially seeing the material.

This would be because I was learning math and could, naturally, not do it very well. Nowadays, I see such proofs as entirely trivial and have no idea what was so hard for my past self to see in this. There's something intangible at work here and when you're just starting it can be easy to see these objective things - like other people being so much quicker to the proof than you - and to not know what you're missing, but, with practice in mathematics, it may come. If you like math, there's no reason to stop now.