Comment Re:Obvious (Score 1) 636
Also arguably, this was more useful to me than rote-learning the proof of the quadratic formula.
I would like to hear that argument.
Like you already hinted yourself, for many people it really is much more useful to devise and implement a clever 'cheat' than to just blindly repeat some mathematical trick the way the teacher wants you to repeat it. There's all kinds of real-world skills involved cheating your way through the common math exercises taught in schools, which usually teach students no more than how they have to jump through hoops.
As an example: myself, I have never been a big star on maths in school, because I found deriving stuff by hand or simplifying goniometric expressions to be extremely boring and pointless. I did (and do) understand the fundamental concepts behind them, but I didn't (and do not) know from the top of my head what the derivative of x^2*log(2x+1) is, and if I ever need to know, I'll just fire up Mathematica. Because I consistently slacked on my math exercises and homework, I got low grades and no satisfaction from mathematics at all.
Now, eventually I chose to go to college and study Computer Science at a university famous for their extremely theoretical, formal CS curriculum, I got my MSc without any major problems, and I got a job writing very complex and very specific simulation models full of FFTs, (non-) linear regression, computational geometry, curve fitting etc. Turned out that skills like being able to find and understand relevant literature, decomposition of problems, and thinking in abstractions, and most importantly: communicating with people who are experts on the theory, are much more important than learning a few tricks so you can show the teacher how to 'prove the quadratic formula' (read: repeat some symbolic gibberish you might not even understand).
If everyone was destined to be a mathematician or a teacher, learning all these tricks and details might be the most useful way to teach mathematics. In reality, most people will only ever need to be able to understand the basic theory behind math concepts, and asking them to prove all kinds of random stuff, derive expressions by hand, rewrite and simplify expressions, it does not help them at all, and it is more likely to scare them away from mathematics completely.