Audio problem: In the early classroom scene, after Judy reads the grammar exercise, Miss Canfield calls on Whitey Whitney, who has raised his hand. When she does, she seems to say, "Harold." (00:01:20)
Other mistake: The opening credits for season 5 show June coming out of the front door with a pitcher of what looks like iced tea, and calling for Ward, Wally and Beaver who are all working outside. But the door opens by itself as she comes out, and then closes by itself behind her. If the entire family is outside, who opened and closed the door?
June Cleaver: Dear, do you think all parents have this much trouble?
Ward Cleaver: No, just parents with children.
Captain Jack - S1-E3
Trivia: The episode "Captain Jack" was the first time a toilet was shown on TV; the boys buy a baby alligator and Wally is shown putting it in the toilet tank. The network censors were so spooked by the thought of showing a toilet on television that they delayed the episode. And when the censors finally relented, the compromise allowed only the toilet tank to be shown, not the bowl.