Both answers are either incorrect or inapplicable to a large portion of the population. The real answer has always been that you need to learn intuitively what

Both answers are either incorrect or inapplicable to a large portion of the population. The real answer has always been that you need to learn intuitively what "division" actually means, and we have no clue how to teach that, but doing it by hand seems better than the alternatives. I'm not sure if teachers just choose not to phrase it this way or they don't know either (in which case the real reason is because they were told not to allow calculators). But calculations are easy. Intuition is hard. Until we have better ways to teach intuition, I'm afraid we'll have to keep trying to let kids build it via forcing a slow-down of the thought process by having them do it by hand. A better example would be integrals and antiderivatives. One can certainly learn how to "find the integral" just by using antiderivatives. One could learn how it applies to the real world, as well. "A ball is moving at 30m/s for 10 seconds, what distance did it travel?" Increment the exponent, divide the coefficient, plug in X. But then kids approach, say, revolutions of solids. Suddenly, due to the way they're taught, they think that there are different formulae for rotating about the x and y axes. If they're taught how to construct the Riemann summation from scratch, see the relationship between that and integrals, and see the relationship between a curve and its revolution, I think that's when they start to get the idea of what an integral (not an antiderivative) really is. It's the difference between saying: I want to find the volume of y = x^2 rotated about the x axis. I have the function and I know I need to multiply it by the formula for a circle and then integrate it. vs I want to find the volume of something. I can describe each point on the cross section. Each item in the summation can be described as circular. I can describe all of these circular sections at once by adding them all together. The integral is that sum. I can find that integral by finding the antiderivative of the original description. Which seems more "true?" Which is more difficult to teach? If kids always used calculators to pursue question 1, they'd never arrive at the problems that would let them arrive at question 2.