As someone with two degrees in mathematics, I think the use of calculators in the classroom is a good idea ("Counting Too Much on Calculators," Sept. 2), both to perform lengthy calculations and to give students some hands-on experience.
At the same time, I am glad that I learned my arithmetic the old-fashioned way, including my father showing me how to do square roots with pencil and paper (rarely taught anymore).
I had a great middle-school teacher who showed us an important connection: Whenever one does "long multiplication" or "long division" or adds fractions with a common denominator, one is using the distributive law, which is basic to a later understanding of algebra. The different arithmetic procedures are then seen to be united, and algebra is seen to "grow out of" arithmetic.
From this law one can solve algebraic equations, show that negative times negative must be positive, and even see why the pencil-and-paper square root procedure works!