Soak black beans overnight. Drain and rinse beans. Place in a pan, cover with water, and simmer on low for an hour.
Drain beans again.
Place all ingredients except cornstarch in the slow cooker.
Cook on high for 4 hours. (Erin's note here: From what I know about slow cooking beans, it won't hurt to cook them longer...say if you'll be at work all day. Cook them longer, then proceed with directions below.)
After 4 hours, use a slotted spoon to remove 1 cup of cooked beans and ham. Place them in a blender or food processor with a little bit of the broth. Puree, and return to the pot. Whisk pureed bean mixture into the broth.
Combine cornstarch with cold water in a small bowl. Mix. Add to the slow cooker. Mix with a whisk.
Let cook for an additional 30 minutes or until broth thickens a bit.