A lot of intersections have "Push To Cross" buttons for pedestrians trying to get through them. At a few of them, I've noticed the signals don't change if you don't push the button. So if you don't bother to push the button, the little white man won't appear and you're stuck with the orange hand, with no indication of when to cross.

But at a lot of intersections, the walk/don't walk signals toggle even if you don't push the button. And at all intersections, whether or not the signals change, pushing the button seems to have no effect on the timing of the traffic lights. It doesn't speed things up.

If that's true, why put them there?

I think it's because people need to feel that they have something they can control. More often than not, reality is beyond our grasp and we're at the whims of what life throws at us. We can't get across the street any faster, but we can pound the button with our finger, and then it feels faster.

Giving people the feeling of control might be just as important, or useful, as actually giving them control.