What are these?

The girls were running around under an oak tree in my backyard. They were playing tag with something, green. When one grabbed the thing away, the other picked up another from the ground and ran again.
Curiosity got the best of me and I found loads of opened green, round things on the ground. Upon further inspection, I saw many more still on the tree branches. I guess they are some sort of pod, but in the 6 years I have lived here, I have never seen them before.