I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was not the best. (I am always too lazy, and not that smart enough)
However time has changed. Now I am dealing with abstract ideas of math, and I realized that I am really depending on visualization. For example, when compactness was first introduced to me, I have no idea about it because my instructor only wrote the definition, so did Rudin. I had trouble understanding the finite subcover of infinite cover because I was thinking "Why not always making the cover be the one covering everything? It is finite." Then I searched a Youtube video and saw a professor drawing circles (open cover) and I was like "Aha! Now I understood it!" I had a period not knowing what cosets are until I realized that I can think of partitions. Till now, after the idea has been introduced for 2 months, I still think of partitions first.
The problem is that very often before I realize what a concept "looks like", I have a hard time understanding the definition/theorem. Often some visualization understanding of a concept come up a period later after it is introduced. During this period, the instructor will go on the study of this concept, and I will be completely clueless. I probably will see why some properties or theorem introduced are true by directly looking at criteria in the definition, say. However I will soon forget without understanding, and such understanding usually is a visualized one.
I don't know whether my method is good. This method somehow looks reasonable but somehow it is like the way to deal with elementary math, but not high-level math. Please help. I've searched some similar topics on this site, many of them are closed because they are too broad or there are too many answers. I hope this one is good enough. I tried my best to make it not too broad. Also please pardon for my bad English, which is my second language.