Your metaphor works both ways. If you teach the kids in kindergarden the theory of intricate japanese wood joints before they know which end of the saw to hold you are wasting your (and their) time.
As I said, all things have their place, but some mathematicians have the habbit of starting at the most general (and therefore most abstract and most distant from the layperson) point to then move to more concrete applications. My suggestion wasn't to skip the general perspective, it was to teach it at a point where people already know and maybe even use the thing that is being generalized.
That being said, not everybody is a mathematician. For some (and I'd argue: most) people using the fourier transformation as "just" a means to figure out the partials of any given signal is the tool they are looking for.
As I said, all things have their place, but some mathematicians have the habbit of starting at the most general (and therefore most abstract and most distant from the layperson) point to then move to more concrete applications. My suggestion wasn't to skip the general perspective, it was to teach it at a point where people already know and maybe even use the thing that is being generalized.
That being said, not everybody is a mathematician. For some (and I'd argue: most) people using the fourier transformation as "just" a means to figure out the partials of any given signal is the tool they are looking for.