Two of your three examples consist of algorithms that take a large number of steps. Without an ability to perform a large number of steps in a short amount of t

Two of your three examples consist of algorithms that take a large number of steps. Without an ability to perform a large number of steps in a short amount of time, there wouldn't be the need to think about such things. That is, your tutoring of the ancient would consist of explaining an algorithm that might take 100,000 calculations. He would grasp it, but shrug and say "so what? It would take years to perform that algorithms. Let me show you a different way that results in a damn good approximation that requires nothing more than a compass, straightedge, stylus and a piece of string." In short, you are confusing reasoning ability with algorithms designed for a particular form of technology. Take neural nets. Would someone from the 1970s understand the benefits of a convoluted net vs. a simpler form? Perhaps. But, without the technology to perform millions of training calculations, the lack of comprehension would come from your pupil wondering what the point would be of trying to understand an algorithm that, with his technology, could never be demonstrated, used, or tested.