What explains a 3-inch diameter circular patch of bad dandruff on the crown of my 9-year-old daughter's head that will not go away?

Tinea = ringworm. That could be 2 or 3 things, but most likely tinea capitis, aka ringworm. It's not a real worm, but a fungal infection. It's somewhat contagious, but won't go away until properly treated. She'll need to be checked out to get that done.
Ring worm. It could be more than dandruff. Your child should be seen by his doctor. Could be tines capitis/ringworm which would require medication to clear it.
Fungal infection. She may have a fungal infection often known as ringworm on her scalp. This can often be treated topically or with frequent exposure to uv light by a medical specialist.