I've always found the Monty Hall problem a poor example to teach with, because the "wrong" answer is only wrong if you make some (often unarticulated) assumptions.

There are reasonable alternative interpretations in which the generally accepted answer ("always switch") is demonstrably false.

This problem is exacerbated (perhaps specific to) those who have no idea who "Monty Hall" was and what the game show(?) was... as best I can tell the unarticulated assumption is axiomatic in the original context(?).