Quote Originally Posted by Alric View Post
you regularly spend fifteen minutes measuring the sides and angles and doing trig to figure it out?
...what does this even mean?

Anyway, you've chosen an extremely basic example. Pythagoras's theorem is so well known it's pop culture, probably only E = mc^2 tops it. Let's try some proper maths; the stuff that you need to learn for exams. How about sin^2 + cos^2 = 1. Do you seriously remember that, symbol by symbol, without reference to anything else? You think that's sensible? Do you also memorise tan^2 + 1 = sec^2, and 1 + cot^2 = csc^2, separately? 1/2*a*b*sinC? sinA/a = sinB/b = sinC/c? All of these are extremely trivial facts if you understand the basis of them.

Even on the most basic level this doesn't work. Exponentiation; do you learn

x^a*x^b = x^a+b,
x^a/x^b = x^a-b,
(x^a)^b = x^a*b
x^-a = 1/x^a

et al? Or do you simply learn that x^a means x*x*...*x, a times? The rest follows easily with a good understanding of this.

Or fractions, even. a/b + c/d. Did you learn that you calculate it via a*d + b*c / bd ? Or do you think it's a better idea to just understand the basics of arithmetic; if you want to add up two things you just have to make them of the same type by changing the bottoms, and you can do that easily by timesing top and bottom by the same number.

Learning reasons rather than by rote is infinitely better: it is self-checking; the knowledge lasts a lifetime rather than a few days; having learnt the reasons behind them, things become far easier to recall anyway; it makes you vastly more confident and fluent; it means can adapt things to new situations with ease.

The entire process of learning in the first place becomes easier, too. Maths is fundamentally a highly interconnected structure. If you understand what went before, learning something new is like driving to a house and placing down some stepping stones up to it; if you don't, however, it's like ignoring the existing infrastructure and building a whole new road just to get home. And then doing this for every single house in the country.

And if those reasons aren't enough, the method is tried and tested by yours truly, and it's working pretty damn well so far. As for all the guys at school who used to spend revision time writing out hundreds of little squares of paper... well, they make good coffee.