Students should not assume that all mathematicians try to learn all the material thoroughly.<p>Instead, in practice, even among good mathematicians, there is a fairly wide range of how carefully they study and how well they learn some material.<p>So, it's possible and common (1) to get mostly just an overview, and even the overview can be at various levels of thoroughness, (2) try to get the main ideas of the most important points, (3) think about the material mostly just intuitively to build good intuitive models that can be the basis of more in learning, applications, research, (4) deliberately go over the material more than once with only the later passes quite thorough. In short there is more than one way to slice an onion.<p>Here is what did me the most good: First get an overview, i.e., what is the material really <i>about</i>? Second understand the details, say, after reading a definition, theorem, or proof, be able to write it down. Third, look back and get a relatively succinct, intuitive overview, <i>model</i>, that keeps all or nearly all the important content.<p>Uh, of the five Ph.D. qualifying exams, I got the best in the class on four of them. For my research, (a) for a paper I published and (b) for my dissertation, I did all the work with essentially no faculty direction. For the research, sure, needed to understand enough low level details of some material, but the real key was intuitive models that led to, permitted guessing, the original math with theorems and proofs.