Perhaps we need to memorize just a few prime numbers multiplication tables (2, 3, 5, 7, 11, 13, 17). We can work out the rest as needed.

One arithmatic process that I think should be changed -- finding square root by long division as taught in primary schools. The method is cumbersome, comfusing and "inefficient" compare to the ancient "Babylonian method" (บาบิโลเนียน see wikipedia or search "computing square root, babylonian method").

This Babylonian method is based on convergence of approimations -- a far more powerful and useful concept for "learning" than the long division method.