I have never tried it myself in a structured setting, such as a classroom; but I do sometimes notice things, and then ask myself, "What is going on here ? Why does this thing behave in the way that it does ?". Sometimes I think about it for a while, figure out what sounds like a good answer, then go on with my day. Sometimes I shrug and forget about it. Sometimes -- very rarely -- I'm interested enough to launch a more thorough investigation. I imagine that if I set myself an actual goal to "observe" stuff, I'd notice a lot more stuff, and spend much more time on investigating it.

You say that, in such a situation, you could end up "feeling tricked", but this assumes that the teacher who told you to "observe" is being dishonest: he's not interested in your observations, he's just interested in pushing his favorite philosophy onto you. This may or may not be the case with Karl Popper, but observations are valuable (and, IMO, fun) regardless.