We all know confidently incorrect people. People displaying dunning-kruger. The majority of those people have low education and without someone giving them objectively true feedback on their opinions through their developmental years, they start to believe everything they think is true even without evidence.
Memorizing facts, dates, and formulas aren’t what necessarily makes someone intelligent. It’s the ability to second guess yourself and have an appropriate amount of confidence relative to your knowledge that is a sign of intelligence.
I could be wrong though.
Memorization is such a strange thing to try to teach. I was never good at that in school but could sing any song on the radio after hearing it once. In school I was good enough at math up to a point because I was so bad at memorization got good at thinking my way through it. Was much better at word problems than equations in elementary school.
I do agree with the premise of the shower thought - part of being educated is learning through mistakes. Making mistakes is one of the fastest ways to learn something, and is the main reason I’m good at my job. I am happy to work in accounting where mistakes don’t kill anyone.