I had that argument given to me at school. In practice it is rubbish.
Just buy a bloody calculator.
I would say knowing stuff like long division is valuable, however I do not know it off the top of my head and have never needed to do it since school.
If I had to do it now I'd figure out how it worked from what I remember and *then* do the problem.
I never actually learnt multiplication tables either. Never needed them.
As a programmer who uses math a reasonable amount (nothing scientific), funnily enough I always have a calculating instrument around.
It is most important to know what tool to use to accomplish a goal, not to memorise the exact workings of it.
E.g. Most people can read a clock, but few know exactly how the clock works or how to make one.