Multiplication:

I’ve much enjoyed the discussion about the multiplication posts, and I agree that calculating in one’s head is a good skill to have. Even if there’s a calculator handy, people are often reluctant to use it (perhaps overestimating the effort it would take, or not wanting to look ignorant or skeptical), and if one can do even approximate calculations in one’s head, one can often spot errors that one would have otherwise missed. Also, my sense is that learning arithmetic is good mental training for other thought processes, but I may be mistaken on that.

Nonetheless, my point was different: The important thing is not knowing how to multiply, but what, when, and why to multiply. Once you understand what calculation to do to check something, you can do it easily enough on a calculator. But the calculator won’t explain to you how to structure the calculation, or whether a calculation ought to be done.

That, incidentally, is one reason that I’m so frustrated when people say “I never minded math” (usually referring to arithmetic or at most algebra) “in school, but I just never liked the word problems.” All of life is word problems! A problem in the real world never comes to you as “figure out 123 x 456 + 789.” Understanding how to translate the real-world condition into numbers is the important skill, not the calculation. And while most school-level word problems are usually only a very first step in that direction, if people can’t master that first step, their knowledge of raw arithmetic or algebra won’t help them actually deal with the world.