When we learn mathematics while young, we are taught numbers using pictures and other kinds of visual devices. That is what helps us learn those concepts. Had they been taught as axiomatic constructs, children would find it exceedingly difficult to learn them. This is not substantiated, but something I truly feel.
The things that I truly feel I understand well are things that I learn from pictures and images. Every picture carries a thousand words (an overused cliche, but true all the same). More so, these pictures are easy to recall later, as compared to mathematical ideas in written formalized form.
My new pet project is to convert mathematical definitions, ideas and proofs into images that I will construct in Microsoft paint or elsewhere, and them embed in this blog.
I remember reading an article by a Fields medallist recently, who described how he learned mathematics. He said he used to write and rewrite proofs and ideas a hundred times over, until he had memorized and internalized them well enough for future use.
With all due respect to him, I hope that the visual conveyance of ideas gets this done much quicker than that. I strongly believe in this project, and shall spend copious amounts of time constructing tell-tale images. I hope they are of help in attracting the best minds of the world to Mathematics.
Note:- I was talking to a bachelors student from IIT Bombay yesterday, clearly one of the most competitive colleges to get into in the world! He sounded really intelligent and well-informed, although that is difficult to judge on a first meeting. He too was interested in pursuing mathematics, having qualified for the olympiad program two times. However, he soon lost interest. He couldn’t understand what was going on in things like Topology and Group Theory. He worked relentlessly, and also took advanced classes like Algebraic Geometry, which he aced. However, he confessed that he couldn’t understand or recall much. Hence, we switched to combinatorics. I also know IMO medallists who feel the same way and eventually shift to something more intuitive like combinatorics.One of them is currently attending the VSRP program with me.
I remember feeling lost too for a long time. Although typing my thoughts down would help me understand the proofs and concepts involved, retention and developing an intuitive feel for said concepts for future use was not up to the mark.
I have tried the concept of visualization- the construction of images has helped me tremendously. Although this project may be huge and monumental, I feel it will be worth it, at the end.