Regarding calculators, they only need a little bit of regulation to be used effectively. They make working with large, small, and complicated numbers easier. In chemistry, for example, I just isolate the variable and let the calculator sort out the mess on the right side. There's no thought in just grinding out long multiplication problems. However, in math, I see a perversion of the subject in the way that we test for knowledge. As you said, checking is of the utmost importance, but we are only given enough time to go through the problem once. If we teach it that way, then 'being thorough' is replaced with 'being a calculator' because the only way that a human can make sure that the answer is right is by plugging the answer back in, while a calculator will always produce the same answer (garbage in, garbage out still applies).
Time pressure causes stress, which causes errors, which misrepresents actual comprehension of the material. Furthermore, 'speed' is not something that needs to be drilled. It comes with experience, which can be gained by working with higher-level formulas right away. Think of it like using a hammer; there's no point in spending hours perfecting your swing, just build something that uses the hammer a lot. The hammer can be any basic idea, and the thing being built using it is a higher-level idea that incorporates it. We do it that way in chemistry. We are given the periodic table and that of the cations and anions, and don't need to memorize it. However, by constant use in many areas I have ended up memorizing the atomic numbers, masses, families, and positions of a good number of elements. Drilling that wouldn't have sped me up that much, and the time was spent on learning more subjects in greater depth.
Do you guys agree that incorporating basic concepts into higher ones is a better option than drilling them for subjects like math and chemistry?
-Penguin
