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Well... I'm an engineering student who intends to graduate in electronics. I became interested in AI when I started learning programming at the age of 12. I became fascinated with what I could make the computer do. And rather naively I tried for months and months to program something that was "intelligent" (and failed horribly of course). I set that project aside temporarily but never stopped thinking about it. Years later I discovered HPMoR and through it LessWrong and suddenly found a whole community of people interested in AI and similar things. That was also about the same time I became a full blown atheist. So while exploring this website, I felt like a kid in a candy store.
Once I graduate I really hope I can do some research in AI.
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So there's an argument for going into the classroom (as a teacher) completely unprepared, stumble through the material, reason things out in fron of the students, go down hopeless calculations for a while, then say "scratch that", "let's see....hmmmmm", etc....Nothing should be clear, the students would have to make huge efforts just to find out what's on the hw. I know some people like this (not by design). I wonder if their students learn some important life-skills though.
I sort of unintentionally had this happen to me. When I was 13 I moved to a new school which was a little less competent than my previous school. In my previous school all the information was packaged and delivered to us to memorize. In this new school the teachers would just roughly go over the topic and it required a lot of independent effort on my part to understand the subject matter.
This led to this mini explosion of clarity in my head. While in my previous school I was bored of learning and depended almost exclusively on rote memorization, this new school forced me to think about why I know what I know. This transformed me into an incredible math/science nut. Just taking the information into your own hands and thinking about it turned me into someone who constantly raved about the beauty of mathematics.