[nhaehnle]

The way people are taught mathematics is mostly useless. It's a bunch of equation and calculation that people are doing by hand. Nobody is trying to identify problems, break them down, make hypothesis, and so on.

I recall some episodes of five minutes wasted by research mathematicians because they were unable to quickly multiply some 2x2 matrices in their heads.

Being able to perform the low-level skills like arithmetic very rapidly in your head helps tremendously when it comes to narrowing down hypotheses. You can often reject a hypothesis quickly if you are fast enough at doing some computations in your head.

So yes, more teaching of how to identify problems and break them down is a good idea. But you also need to be rock-solid in those elementary but fundamental skills, and you don't get there without a lot of practice.