The key is "you need to play with it". The way to get this to work in practice with elementary mathematics is something like Guesstimation. To my knowledge, thi

The key is "you need to play with it". The way to get this to work in practice with elementary mathematics is something like Guesstimation. To my knowledge, this is not taught in most math classes. And the reason is simple: it is messy. They might try something that looks similar with word problems, but that is a far cry from the majesty of investigating the world with simple arithmetic. The problems they come up are contrived and students are no fools. Problems must come from within, not be handed to people. Drilling vs understanding is not an either..or. Most learning starts with messing around, trying to figure something out. It is slow and will be painful if the interest is not there. At some point, understanding is sufficient with a foundation in place and then drilling comes in to get a mastery, but what "drilling" is and needs to be will vary from person to person. Too little, and the speed fails to materialize inhibiting understanding of higher level concepts. Too much and it gets boring shutting off critical interest. All of this suggests that teaching mathematics in a one-size-fits-all is difficult. Also, imagine trying to grade 120 guesstimations for weeks on questions the students come up with. This is the essential difficulty of our current teaching mindset, namely the single authoritarian figure blessing or cursing the work of others. The key to true learning is excitement and motivation and little external judgement. Interested people will largely self-correct with a few subtle pointers here and there. Even at the level of Fourier transforms, most students probably fail to understand the problem. So how can it possibly make sense? Math is about working around hard problems, but the teaching of mathematics leads one to believe that there are simple algorithms to apply and we are done. A lower level example of this is Newton's method. Solving f(x) = 0 in their experience (quadratic formula) is easy and graphing shows the zeros anyway. What is the problem it solves? But what if you just give them a graphical piece of the curve and ask where they think a zero is? Then the problem becomes clear and the solution can make sense. Then it gets translated into steps (draw a line that fits well, find its root, find out what it looks like around there, repeat). And then to master it, one can do a number of drills. Good luck doing this, however, with people who do not care one iota about it and live in a culture where the majority of Americans have disdain for math (probably born out of a mixture of shame of how poorly they did in math class and bitterness that this most wonderfully human tool for exploring the world has been denied them).