Students definitely should doubt their teachers ---- who often make errors, and doing so should inspire more learning. If it gets out of hand, they can always be given a detention for it.
There is a place and time for shoveling as much information into a child's head as it can possibly hold without exploding. This is when we teach multiplication tables, drill grammar into their thick skulls, teach them basic math up through algebra, spelling, penmanship, history, and so on.
These are unnecessary. Strong problem solving and critical thinking are more important in life and should be learned first.