It is something of a problem in the teaching of the early stages of math, as opposed to arithmetic. Most people like to have motivating examples, most people l

It is something of a problem in the teaching of the early stages of math, as opposed to arithmetic. Most people like to have motivating examples, most people like to see that what they are doing is relevant, and that it makes sense. Trying to tell people that F=>X is true, regardless of X, offends most people's sense of what's reasonable. Consider. Suppose that in a tennis match each point is won or lost independently of all other points, and suppose further that the server has a 95% chance of winning each point. It seems reasonable to assume that the probability of winning the game from a winning position is always greater than the probability of winning the game from the beginning. But this is not true! The probability of winning the game from 40:30 up is less than the probability of winning from Love All (99.986% vs 99.991%). Some students, even those who are quite competent at probability and can follow the calculations, find that this offends their sensibilities and decide that it's all rubbish. People get very upset if you contradict their common sense. To claim that the sentence: If n is an integer and both even and odd, then n=17. is true gets some people really confused. The hurdle of understanding why it's true needs to be passed, otherwise most of math will remain unattainable. (edited to add the actual probabilities)