Pencil-and-paper arithmetic: are kids still supposed to be able to do it?

In My Humble Opinion
41

Keep in mind that, even in 1869, nobody is going to calcuate the cube root of 9.3 to five digits by hand, without consulting a table of logarithms. I tried it and it is… not so difficult, yet time-consuming and the chances of making a mistake begin to approach 100%.

So, why is it on the test? The answer is that it is a pure exercise supposed to test that the kids know what they are doing and can come up with and apply an appropriate method. Which brings up the danger that they memorized some algorithm without really understanding what is going on. [Feynman recounts a relevant anecdote.] Even today, it is up to the test writer to devise probing questions even while allowing them to bring along calculators and computers and what not.

42

Some years back, I was proud of myself for developing a method for approximating square roots that requires only additions and multiplications, aside from a single division at the end. I later learned that the method goes way back, and was possibly even used by Archimedes. Key to the method is that, given two fractions \frac{a}{b} and \frac{c}{d}, the fraction \frac{a+c}{b+d} is always somewhere in between the two.