[yequalsx]

Using word problems is one likely to discover why sqrt(2x+1) = sqrt(x) + 1 is fundamentally different than cuberoot(2x+1) = cuberoot(x) + 1? Give this problem to calc one students and many will do both problems incorrectly. They'll incorrectly cube both sides in the second equation. They don't understand why we solve sqrt problems but not cube root problems but they know how to compute doubling time for bacteria growing in a petri dish.

Your fourth paragraph is an argument in favor of what I've been saying. The applications we teach ought to be applications to math; not to physics or biology.

But back to the apathy problem. People aren't going to be motivated by number theory or bacteria in a petri dish if they possess too great a disdain for knowledge.