Cognitive dissonance has interested me for many years. Perhaps my favorite illustration of it is the classic mathematics “Monty Hall Problem”: A grand prize is hidden behind one of three curtains. After you pick a curtain, the host shows you one of the other two curtains which would’ve been a loss. You’re then given the option of keeping the same curtain or switching your choice to the remaining, unrevealed curtain. What do you do? Spoiler alert! The next paragraph has the answer.
While changing curtains has a 1-in-2 chance of getting you the grand prize, keeping the old curtain had merely a 1-in-3 chance because it was chosen when three curtains were still available. The fact that we now know one of the choices that would’ve been a “loss” doesn’t impact the probability.
This problem took me awhile to comprehend when I first saw it. The same thing happened when I posed this problem to a mix of 7th and 8th graders. “But it’s a 50-50 chance either way!” they protested. Only after collecting data simulating the experience over 50 times did they start to come around to the idea, and even then it was usually, “Well, that’s probably right but I still don’t believe it.”
Cognitive dissonance may also describe the larger view our society hold of mathematics these days. According to this great interview with mathematician and computer programmer Keith Devlin, the use of pencil-and-paper algorithms is largely grounded in a 19th-century world. It took me a minute to catch that this didn’t even mean the 1900’s, but rather the 1800’s!