Are you sure you didn't mean to post this in the "Stupid Questions" thread...?
Sure, I don't remember my chemistry too well, but particles don't just "fall out" out of solution. If a solution is saturated and extra particles are added, particles will constantly dissolve/un-dissolve (yay unscientific terms) while an equilibrium is attempted to be reached. Particles don't just reform on the bottom of the cup and stay there, they immediately dissolve again while other particles fall out of solution.
As for why stirring helps dissolve the particles on the bottom of the cup, stirring increases the surface area available for the solution to dissolve the particles. It's easier to dissolve a sugar cube by submerging it in water than it is to pour water on it.
Or something... Like I said, it's been awhile.
I'd provide a drawing, but what am I? A science textbook?
Point is, you're probably just hitting the bottom of the cup in different places or with different parts of the spoon. Or you're just crazy.