I agree with you absolutely on breaking up the problem. Math is my baby, and what you say is key. Math is incredibly simple..and hard. The problem is that our brains are wired to understand certain complex tasks, not simple ones. We can understand speech, recognize voices and faces, complex tasks all - but cannot reliably add 1000 numbers together even though that is simple.

Being good at math is largely a process of understanding this, taking things in pieces, and breaking them into appropriate pieces if they are not that way already.

Compared to math, programming is a nightmare.

Programming involves dealing with a machine that needs the same sort of approach, however it also involves dealing with users who won't specify problems, and a real world where there are many ways to break problems apart. Choosing which one will work well in a given situation is key. It would be easier if technology stayed still for a while, but it does not.

Now you are right that around here we deal more with the mechanics of programming rather than the hows and whys. Unfortunately the hows and whys are going to age better than the mechanics. And I don't know how to communicate them. I can show the ones that I know, but someone who does not already understand it will miss the point. And it is certainly possible (in fact I can think of examples here) to master all of the mechanics without having a clue about what I would call "good taste"...