Has Science Disproven Free Will?

Occasionally I read - usually at Jerry Coyne's blog - of some study that is supposed to cast doubt on or disprove free will.  The study usually involves hooking up the human subjects to machines that scan their brains while they are asked to make some arbitrary choice.  Then the machine detects that the brains have made a choice before the subjects are consciously aware of making it themselves.   "Aha!" say the researchers,  "This proves that the choices were physically determined by the brain, not by the conscious decisions of the subjects."   Then someone like Coyne comes along and generalizes from these studies that all of our conscious decisions are physically determined prior to our consciously choosing them.

Perhaps people like Coyne are right.  But before we swallow the argument, let's chew it over a bit.  If I'm asked to make an arbitrary decision, one which I can find no conscious reason for deciding one way or the other, how do I go about making my choice?  From my own instrospections, I seem almost to pull a lever in my brain, as if it were a slot machine at a casino, and let it determine what choice I should make.  Then the brain seems to "spin" for a short period of time until it determines what alternative I should choose.  At some point in time I consciously make the choice.  But it wouldn't surprise me to find out that my brain had already made the choice before I consciously knew about it.  So it doesn't surprise me when studies suggest that this is what happens when other people are asked to make an arbitrary decision.  We don't have any conscious reason to choose one alternative over another, so we let our brains do the choosing for us.

What I think would be a more interesting study is asking people to do a logic or math problem and then choose what they think is the right answer.  I'm curious what the brain machines would show then.  I suspect that the conscious choice of an answer would be either simultaneous to or maybe even prior to what the brain chooses.

But perhaps such studies have already been done.  If so, I would be curious to know what the results were.