African Violet Growing Long Brown Stems?

I have several African violets that are healthy and bloom a great deal. Their heads (made up of leaves and flowers) are very large. They are also outgrowing their pots. Two of the larger plants look like they are growing over the sides of their pots on large stems. These "stems" are brown, scaly, about the size of a woman's little finger, and have what looks like short, brown, dried roots extending from them. I have been very careful so as not to break them, but I have never seen anything like them in all the years I have grown violets.

If I transplant them, what to I do with these stems (except pray not to break the head of the plant off the stems.) Do I plant these stems into the soil so they are covered? Or should I find a really large pot and plant the main roots and the stem in the middle of the pot and let them continue their growth track over towards the side of the pot.

None of my readings on violets mention this type of problem.

By NM