In the military, I was a Field Artillery fire direction chief. Over the course of my career, we moved from manual, slide-rule gunnery to various stages of electronic ballistics computing. But through it all, we still understood the manual basics. If you don't understand that, then the computer makes no sense, and when it goes wrong, you don't know why.
I think mathematics is a lot like this. Do we rely too much on our computers in our United States math classes? This guy seems to think so:
It reminded me very much of a conversation that I had a few months ago. The person I was speaking with was a programmer visiting the U.S. from India. This gentleman and I had gotten to know each other fairly well, and he has been programming for quite some time, probably around as long as I have. He and I got onto the topic of education. It is a well-known fact that India, China, and a number of other countries are beating the U.S. in math scores. So I asked him how they teach math in India. He was almost baffled by the question, as if there was more than one way to learn math and this was the first time someone had let him know. “From a book, with examples on the blackboard, how else?” I queried him about the use of calculators and computers, two tools quite common in U.S. math education. He explained to me that calculators are forbidden in their version of high school and that, in colleges, the calculators allowed are basic models (think add, subtract, multiply, divide, exponents, square root, log 10, and natural log), and their usage even then is frowned upon to the point where using a calculator will be the cause of ridicule and humiliation. This sure sounded like a far cry from the educational environment in the U.S., where 7th grade students are now being required to own TI-85’s, a calculator that’s probably more powerful than the guidance system on a cruise missile.