Bring to a boil, and cook until potatoes are soft enough to break with spoon. Stir often. After potatoes soften, reduce heat and cover, and simmer for a while, to allow flavors to blend; about an hour or so. Keep a close eye on it.
*This is the one-pot method; but you can cook the potatoes in a separate pot until soft, and meanwhile cook the hamburger/onion mixture in a skillet. Once the potatoes are done, drain off some of the water, and add meat/onion mixture, along with the green beans, if desired. Keep in mind you will still need to cook it all together for a while, to allow the meat and onions to flavor the broth. This could also be cooked in a crock pot. We like to eat this with a glass of milk, and bread and butter.