I want to "know math" I've taken calc I,II,II, and linear algebra I, II, but I don't really feel like I get them, even though I got As and Bs in all of them. I feel like my understanding of the meaning of a limit, derivative, matrix, integral, eigenvalue, differential, etc. are all mediocre at best and I only did well on tests because I grinded practice problems for a few days before the tests and memorized the methods of computing things. God help me if you try to get me to explain some of the more esoteric stuff I managed to get a decent mark on like the methods for solving differential equations.

How can I go about gaining a good understanding of concepts? I want to be able to explain what these things are in clear language, I want to be able to build them up in my head from basic algebra. What kind of books should I be reading for this? Since I was not a math major, I just attended lectures and did practice problems trying to get good grades - I never really read any textbooks, will reading them accomplish this goal?