Disclaimer: I am a graduate student about to begin his PhD. I don’t claim that the following method is the “best” way to learn Math with any credible authority. I am only sharing some thoughts that I think have improved my Math, and hope that they can help others in my position too.
I spend an inordinate amount of time thinking about how best to learn Math. It might seem a waste of time to most people, as they might think that I should spend more time doing actual Math than waste time on such pedagogical questions. However, I stuck to this process of introspection for very many years, and it is precisely because of such analyses that I believe I am doing the best Math of my life these days.
Method: Whenever I learn a new concept, I attach a geometric example to it. When studying Algebra, for instance, most of my examples come from Algebraic Geometry. Hence, a prime ideal becomes the set of polynomials disappearing at a point. The Snake’s Lemma becomes the set of polynomials on a line, mapping to the set of polynomials on a point of that line.
Reason for coming up with this method: While thinking about how to best learn new Math a couple of weeks back in the library, I realized that I think much better with pictures. For instance, I would be much more comfortable thinking of a continuous function as a “curve without breaks” as compared to a map between two sets such that the inverse of an open set is an open set.
One may argue that the “line without a break” definition does not capture all the implications and consequences of a function being continuous. A continuous function is a much more general concept that may be used as a map between spaces that the human mind cannot even imagine. One must not lose sight of the beauty and the power of this concept that much smarter mathematicians have contrived over a century.
That person would be absolutely right. However, let me offer a bad analogy: imagine a programmer trying to program. The programmer speaks a certain language that comes more naturally to him, and the computer understands a different language. The compiler translates the language that the programmer is more comfortable with, into the language that the computer can understand. The programmer may not be comfortable using the language that the computer uses.
Similarly, think of formal Math as the language that the computer uses. It is the most powerful language that the human mind has ever come up with, and is the closest approximation to “truth” that we have. However, despite years of trying, I’ve never really been very good at this language. Now I’ve stopped trying. And it is definitely for the best. I have started thinking in elementary visual examples, sacrificing generality for visual intuition. And the process of translating such intuition into formal Math proofs is not very difficult.
Anecdotal evidence: To give you an idea of how much easier this approach has made life for me, let me just say that I am now comfortable with more Algebra, Algebraic Geometry, Algebraic Topology, Differential Geometry and Complex Analysis than I have ever been in my life.
The way Math is generally taught pushes people away from the discipline. I really hope that this post helps people learn Math a little better.
EDIT: New mathematical blog posts coming soon.