Fixed gear wheel NOT stopping when feet stop. Any ideas why? I have a Pure Fix fixed gear bike. I've been riding it in single speed mode for a while but recently decided to flip the wheel and make it a fixed gear. 

My problem is that when I try to stop moving my feet on the pedals, the bike wants to keep moving forward. Like the momentum of the movement is causing it to be extremely hard to stop. The wheel doesn't just stop, and the pedals push back against my feet. That means I'm having to put a great deal of force on the pedals in order for it to just slow down!

Aren't fixed gears supposed to do exactly what your feet do? I used to have an old bike a long time ago and it was really easy to stop the tires from turning, making skidding or braking a breeze.
That bicycle you had long time ago had rear brake (actuated by pedals turning backwards), the one you have now has no rear brake and you have to fight force of inertia with your feet to stop it. One might be tempted to add that this is the exact reason why brakes were invented and added as standard equipment to bicycles.