If you controvert something, you dispute it. If you argue for a new pony, your parents will controvert your argument by pointing out you said the same thing about the dog — the one they have to walk.
The verb controvert can mean to prove to be false or untrue. If you are a historian, you know that new evidence often controverts commonly-held beliefs, so history is constantly being revised and updated. Many children grew up learning that Christopher Columbus landed first in America, but now we have evidence to controvert that idea and other explorers such as Leif Eriksson beat Columbus to America.