Re: When did the multiplication become something to learn by heart and which im

I don't know of any date or place when elegant ad hoc number juggling 
suddenly gave way to formal rote instruction.  Classical aesthetics are 
invariably displaced by efficency.  Verbal problems from ancient Egypt are 
incredibly hard to understand because they didn't have efficient symbology 
to squeeze the goo and dribble out of literary exposition.

Poor Isaac Newton and his "Principia" were mired in continued fractions.  
English mathematician Henry Briggs first introduced decimal notation, the 
decimal-based system of logarithms, and long division in 1617.  They were 
radical and dangerous ideas.  If you think doing long division by hand is 
nasty, look at the alteratives!

Who has to know arithmetic?  As civilizations became more technological 
their participants and geater fractions of their participants needed more 
real time arithmetic skills to be productive.  Personal gain is a powerful 
incentive, more so if you get to keep it. If there are surpluses, somebody 
had to keep track of inventory (soon followed by auditors to watch the 
watchers).  Convenience is important, especially if your number system 
doesn't have a zero (Roman numerals). 

If *you* have a calculator, do *you* need to memorize anything?  The 
understanding required to use a slide rule is non-trivial.  One imagines a 
chimp could be trained symbolically use a calculator.

1) Did you punch the wrong numbers?  Approximation is a useful skill.
2) How many significant figures are justified in your answer?
3) What do you do when the batteries go dead?

If you use money and can't do arithmetic in your head, you will be cheated 
until the lessons are driven home.  Welcome to technological civilization.

Uncle Al!