What we want is like messing about with instruments to see what they do,
    and getting hooked on the curious things that are possible.

In high school, I wanted girls to have sex with me and adults to leave me alone (so I could get better at football and maybe girls would have sex with me). I got good at math to effect the latter, and it so happens that I got quite good. Until calculus senior year, I don't ever remember having any notion of "the curious things that are possible".

Now I work in scientific computing.

Anyways, I don't know if student engagement is as important as you seem to think.