Whether you enjoyed mathematics in K-12 school or not, most people have never been exposed to “real” math. The first phase of mathematics as taught in K-12, and even as taught to engineers and physicists at the university level, is often dubbed “computational” mathematics. This typically involves using established methods to compute numerical results or, later, to manipulate symbols to derive formulas. An example of the latter is finding the derivative of a function in calculus.
For the most part, only students majoring in mathematics (though sometimes also physics and engineering students to some degree at some universities) are exposed to “real” mathematics as practiced by mathematicians. This involves using logic to prove mathematical theorems, and generally has nothing to do with numbers but rather mathematical structures. It requires both an intuitive understanding of the (often abstract) concepts as well as the ability to use logic and proof techniques to create rigorous arguments. This is often called the second phase of mathematics, and is what is typically studied by mathematics majors at the undergraduate level.
I am somewhere in the middle of this phase of my mathematical journey. I had studied plenty of computational math formally in school, and in fact switched to electrical engineering for graduate school (I received my bachelor’s degree in computer engineering) specifically because I perceived it as being more math intensive. However, it did not occur to me until doing my doctorate in engineering that math could be studied for its own sake!
I greatly enjoy this pure math (this is in contrast with applied math, which concerns itself with solving real-world problems as done in physics and engineering) much more than the computational flavor of math. Coming up with a proof feels more like a puzzle or brain teaser than applying the rote methods of computational math. Additionally I find the abstract structures studied in pure math absolutely fascinating in and of themselves.
As I understand it, there is also a third phase of mathematics, which is doing mathematical research. In earlier phases one is often trying to work towards a known result (for example, figuring out how to derive a known formula or prove a known theorem). In research, the end result is not known and is something that must be determined as one proceeds. Though creativity is definitely required in the previous phases, it is most prominent in this research phase. This is what is conducted by mathematics students at the graduate level and is what is done by practicing mathematicians. The results of the activity are usually published in academic journals and expand overall mathematical knowledge.
One thing I have learned since starting to study pure math is that mathematics, like most other fields, is very vast with a lot of disparate topics and sub-topics, and it is not possible to learn all of it in a single lifetime. There is a famous saying that David Hilbert, who died in 1943, was the last man to know all of mathematics. While I cannot say for sure, it is my impression that most of this vast ocean of math is pure math with no current application. This being the case, one could probably spend their entire life studying math at the second phase. I do not know yet whether I will ever reach phase three or whether I even want to because I enjoy learning new math so much and there is so much to learn! For now I am just enjoying the journey.