Our team has had an extraordinary amount of parental participation this year, with two parents coming to almost all the meetings. Seriously, that's a lot. For us.
But as far as team interaction and the balance of power goes, we (the engineers and parents) start out by going over the game with the students, and accentuating those salient features in the game that would lead to the conclusion that we think would work best. But then the students decide on some other way to play the game that they think would be better, and we build that.
Our teachers are, in large part, not that interested in the team, although there are several exceptions.
Some pay no attention to what's going on (40%), others ask us questions and appear interested, but never pick up a tool (50%), and one guy is about as gung ho as it gets. He's a shop teacher, and is very good at manufacturing things that are solid and precise. (As an electrical engineer, my manufacturing skills are somewhat sparse.)
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So it breaks down like this: (assume engineers, parents and interested teachers are lumped together under "engineers" below)
Strategy/Top level design parameters: Students as a whole, with engineers trying to influence, but being unsucessful
Low level design : Subsystem students and engineers (exact mix depends on the subsystem)
Manufacturing: Engineers with student support (NASA safety rules prevent students from operating bench tools)
Integration: Students with engineer support
Driving: Students with engineers cracking the whip (figuratively, of course: it's more of a crop than an actual whip).
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I'm also interested in the breakdown for other teams - please post, everyone!