People value intelligence and will apply that label to themselves even if it may not be true. It's a hard thing to measure though IQ does it reasonably well. To those who have a high IQ: what is it like? Can you pick up any book, read it, and understand the gist with minimal repetition? Can you infer solid and accurate conclusions based on a small amount of evidence? Is any subject or discipline up for grabs or do you have to have a keen interest in a particular field in order to flourish? What is something you are able to do that you know is because of your intelligence -- the proverbial 1,000 pound deadlift of the brain, if you will.
There is no point to these questions other than curiosity.
It's not just "public education", it's been the perception since forever. It's the flip side of the coin to deaf people who can't speak being considered automatically "stupid" (hence the shift in meaning that the word "dumb" originally meant. Medieval writers knew the difference.)
The best example in popular culture is probably the goddamn Professor from Gilligan's Island. The castaways expect him to know everything - but he's a goddamn botanist. No wonder his glue didn't hold - he's not a chemist. He might know THAT a certain sap or plant is used for something, but not how to use or extract what you need from it.
Or Hermione from HP. Great at book learning, but she couldn't get it though her thick skull that House Elves are not slaves and don't want to be "freed". Ron and Hagrid tried to tell her that Dobby was an exception ("There's oddballs in every bunch, Hermione")