The above picture shows my 6 year old making chewing gum. The process to make chewing gum basically involves mixing powdered sugar, flavoring of choice, and a gum base. Here are a couple of websites I found that have more specific instructions.
My daughters had fun making their own gum and then trying it out. They seemed to like it. My wife and I tried it. It was okay for about the first 5 seconds and then the flavor was completely gone and I have to admit it tasted a bit disgusting. LOL! That doesn't, however, mean you shouldn't try to make your own gum at home with your kids. There are many different ways to make gum, so try out a few to see which method works best.