Some key parts:
Some things the brain can do quickly and intuitively, and some things the brain has to emulate using many more of the brain's native operations. Sometimes thinking in metaphors is a good idea, if you're human.
In particular, visualizing things is part of the brain's native architecture, but abstract symbolic manipulation has to be learned. Thus, visualizing mathematics is usually a good idea.
When was the last time you made a sign error?
When was the last time you visualized something upside-down by mistake?
and
The principle of "use the native architecture" extends beyond visualizing mathematics. Back in my senior year of high school, Eliezer once mentioned to me that Chinese speakers were able to memorize longer strings of digits because each digit is a single syllable in Chinese. As a computer programmer, it occurred to me that there was nothing stopping me from picking another encoding - and I have perfect pitch, so I picked musical notes. Middle C is 1, the D above that is 2, and so on up the scale; 0 is the B below Middle C.
Thus, when my psychology teacher put up a string of twenty digits on the board and asked us to memorize them, I was able to do it. In fact, I still know that string of digits, as well as several phone numbers I used this trick on (though I stopped bothering once I got a programmable cellphone).
I have to wonder if this is the real reason how some people manage to do amazingly fast arithmetic - using some sort of native computation, be it calculating emotional states, doing color or pitch transforms, spacial positioning or something else.
