We started telling a story about this group of kids that go on an adventure in a fantasy environment. Alice named them the "Adventure Kids". I carry the majority of plot lines but she has to make decisions for her character and respond to questions and other characters while in character. So the stories are very interactive and anything is possible, to a degree. We roll dice to see what happens when we feel like it but sometimes we get riffing on the story and don't bother.
Alice took to the mutual story telling right away. She loves it. We had some discipline issues initially because she would get upset when we would have to stop for dinner or she would beg for it at all times of the day :-) We got past this quickly and we now have settled into a daily Adventure Kids session starting around 4 pm, usually running for 1 to 2 hours.
Lately she has started trying her hand at carrying the main story line. She'll set up a conflict and have everyone react and discuss. She'll ask me questions about my character and roll, etc. So, lately I've been GM (i.e. Game Master) for one or two encounters and she runs the final encounter.
I love spending time with her doing this. It is tiring keeping up with the pace of the story (we'll switch to pre-made adventures soon) but with her doing more its getting pretty fun. She has creative and funny additions to stories and gets really animated acting out her characters.
This provides us with a great way to be creative together as well as a way to set up ethical dilemmas and talk through them - the game provides consequences to decisions, etc. I'm really liking it as a tool to discuss difficult topics of behavior and leadership. The main thing is that it is easy to keep her attention with the story - she loves it and loves building it as we go
UPDATE: Oh yeah - we also get practice with numbers and subtraction/addition, along with new vocab words. Probability is in there but I haven't started talking about it directly.
UPDATE: We went by Great Hall Games in Austin this Wednesday, after our weekly lunch with Cristy, so she could pick out her own dice and dice bag (she wanted her own girly ones... she was tired of using the ones with "boy" colors :-). She picked out a purple dice bag and has many sparkly dice of different colors. I was so proud of her