[encrux]

I will argue that you don't even need real world application. What I do find really important is motivation of a problem.

Before even talking about math, teachers just need to explain where the problem came from, who was thinking about it and why (if it's known specifically, like cardano's 3rd degree polynomials, even better).

If there's one thing that gets me (and by my own observation, the high school students I tutored) motivated, it's this by far.

[corrral]

Agreed, the motivation doesn't necessarily have to be an application, though that's probably one area worth emphasizing way more than it currently is, especially in secondary education—lower elementary, in particular, doesn't really have to struggle to communicate applications because counting and basic arithmetic are easy to find uses for.