I think that story gets blown way out of proportion. If you picked random people on a street and asked them to integrate 4x³ from 0 to 2, you would maybe get somewhere between 1% and 10% correct answers, probably way closer to 1%. Still this is something that maybe half of the people once learned how to do in school, they just forget it because they don't need it in everyday life. But spend an hour or two and you can certainly teach them once again how to do it. And most of them will of course also forget it again pretty soon. Along the same line it shouldn't be to hard to give someone a rough idea of the difference between classical and quantum mechanics, notwithstanding that most people will pretty soon only have faint memories.