http://www.nytimes.com/roomfordebate/2010/09/26/why-more-students-rely-on-tutors
One of the respondents, Dr Ormsby, opined that we need to get rid of calculators and bring back proof-based geometry in our math curriculum. He is right.
I studied Japanese abacus and mental calculations in elementary school, and indeed abacus and slide rules help students form a physical understanding of arithmatics, an understanding you do not get from calculators. Some kids will get arithmatics without an aid, but most students benefit from this physical understanding. Essentially, you are engaging the visual and the physical centers of the brain in understanding math, in addition to the verbal center. Anytime you use more than one part of the brain, you get better learning results.
In East Asia, elementary school students used to train on the abacuses as well as calculators as part of their arithmatics program. Of course, memorization of the multiplication table was a requirement, too. I don't know if these things are still there today, but I hope so. America would do well to import these requirements.
Proof-based geometry is another vital part of a rigorous high school math program. Students who cannot do proofs are not ready to take on technical majors in college [except maybe biology/psych.] We are doing students a disservice by depriving them of this vital piece of learning, therefore constraining their options in college. Proof-based geometry [and all proofs] is fun, more so than algebra, which is just pure drudgery. If you take the fun out of math and leave only drudgery, of course students will tune out.
When I was doing stuff like math teams, Academic Decathlon, and SATs, I found that I could do the problems more quickly without a calculator, than with one. The abacus training was great in that regard. During my time in AcaDec, the math sections were difficult for most contestants to finish, for some reason [I didn't have a problem.] Maybe it was when they were introducing calculators into the problems, thus causing that problem. Anyway, for contestans like me, it was a great place to pick up valuable points and pull away from the pack. I hone the strategy of quickly deciding if I should apply calculators to a particular problem or go w/ my mental abacus, which allowed me to complete the math sections. For standardized exams like the SATs, the math section is indeed a place where fast arithmatics is vital, and where a calculator will slow you down.
[I was extremely saddened when they subsequently dumbed down and trimmed the math sections at AcaDec. Going from a 50-problem set to a 35- -problem set is just pathetic.]
New York Times had another recent article on the testing culture in East Asia, and it is fascinating. I may have to put my kids into cram schools during their elementary years, just to make up for the lack of structure and discipline of today's American elementary education.
http://www.nytimes.com/2010/09/12/weekinreview/12rosenthal.html?scp=1&sq=chinese%20testing&st=cse
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