The previous post dealt with the question of what maths to learn. This post will deal with how to learn efficiently.
In a casual discussion, Steven Gortler mentioned that books on modern mathematics are so abstract that one may not grasp clearly what the initial questions are. And these books tend to focus on proving theorems (after all that’s what mathematicians do). He specifically referred to differential geometry and topology. Some of the most helpful handouts on differential geometry he gave are actually taken from a physics book on general relativity.
Keep in mind that most mathematicians (and hence maths textbooks) are symbolic types, and that the maths books are really geared towards those types of minds, sifting through different textbooks for the one suitable for geometric types is not an easy job. Luckily, Steven has done the worked for me.
In a larger sense, the books or the lectures presented by a particular type of mathematical minds will likely to strike the right cords of only that particular type of students. While in self-directed learning, I have the luxury of sampling differential textbooks and skipping lectures as I wish, those less fortunate (a.k.a, the students who are actually taking class for credits) are likely to experience not only intellectual hurdles but also confidence problems. I had certainly doubted my mathematical skills after physics classes by theoreticians (strangely as it sounds but true). Having Maxwell articulated the difference between symbolic and the other types proves to be helpful in understanding the difference, finding my own preference, and establishing some criteria for selecting maths textbooks.
Besides textbooks, other tools are also available. I am testing a few maths softwares including mathematica, matlab, and my own rhino scripting.