I think it's relative to the use to which you intend to put that knowledge. There are zillion of different algorithms that could be applied to do long division

I think it's relative to the use to which you intend to put that knowledge. There are zillion of different algorithms that could be applied to do long division. I could for instance, apply an algorithm that works for first order classical logic extended with the set of peano axiom and a successor function. Certainly someone who knows THAT algorithm understands something more about division than folk who just use paper and pen with arabic numerals. But do most people need to see the relationships between the two algorithms to do all the stuff in life that they need to? Clearly not. So to be convinced of your position I'd need to see why the particular knowledge of the particular algorithm you think is so important is really necessary to the work that most people go on to do with long division. I'm not saying the issue is completely cut and dried - but I personally can't see what your argument might be. - edit - I might just add... that it's ALL technological reliance to some degree. I mean - what if they don't have pencil and paper? What if they don't have arabic numerals? These both were important technological advances that made long division as we know it possible. It simply couldn't be done with an abacus in any way that wasn't horribly time consuming. Technological reliance doesn't hold as an argument against employing new and faster algorithms except where it may still be difficult for many to get access to that tech. In the case of calculators in the western world - that's clearly not an issue.