We are super motivated when it comes to learning certain topics in mathematics, because we need the theory to solve our research problems. We need to know if something is true or not to make progress. Most often we anticipate something to be true, because it fits with the net of knowledge. With net of knowledge, one does not care about what implies what without any loop in logic; one only cares about if something is true. If something we need to be true were false, the whole net would seem to be on the brink of collapse. When it comes to writing down our work, we then need to know if a proof is available because we need to provide references for both our potential readers and ourselves. This is where literature comes in. If what we need is fundamental and basic, then the result is in textbooks rather than papers. (These results were first contained papers and when the theory is ripe enough, someone would write a textbook on the theory.) We expect the textbooks to provide terse and exact statements of the results and explain the proofs in an accessible way. We don't want to see a meandering paragraph about motivations. Short ones are OK. (Just tell us how modules over Dedekind domains are like and shut up, for example.) We have our own motivations since we need to use the result in research. This is what researchers want of textbooks. Unfortunately, sometimes people teach directly out of these textbooks. People should teach out of books especially written for teaching purposes. Why are these two different types both called textbooks?

We do need a ton of basic knowledge before being able to do research. It is tricky how to acquire that ton of knowledge. We are those type of people who can gulp down stuff without motivation... We get high simply by seeing dots get connected in our net of knowledge. For those who need motivation, one may not be able to understand the motivation before actually learning the basic knowledge, so don't ask for motivation too early. Sometimes the best way is to follow the book slowly and painstakingly. It takes effort to understand complicated things. Sometimes one is just finding excuses for not learning. Really sometimes it is your problem for not understanding.

It is not true that one needs to have all basic knowledge before starting to do research. One fills the holes in the net of knowledge by doing research. One is driven by the need to know, so one will see what is missing and pick up the knowledge on the way of research. Whenever one learns something, one thinks about connecting it to the research problem, subconsciously...

[Edit: Coming to think about it, most of mathematics cannot be taught. Calculus and Linear Algebra are the only topics in mathematics taught to the general public and people complain a lot as is always the case. People just whine about everything that they are not comfortable with. Sometimes it is really not a problem of teachers; the problem is that most students are too lame for mathematics. We personally are a slow and single-threaded thinker. Our teachers fared much better with us when they just shut up for 5 minutes and let us concentrate to think things through than when they kept explaining and occupied our brain with the task of deciphering streams of sounds. Because of our tendency to block, people don't like to talk to us. ;_; ]


What did we do? We actually commented on doing research in mathematics. ?