Eh, I don’t think they need to be banned. I think their use should be strictly limited.
While you *can* figure out most derivations by hand, it takes forever. Once you’ve gotten the concept and methodology down, it is instructive to be able to quickly show the result of a derivation and how alterations affect them.
As for computers - if it weren’t for the relatively high tech nature of my highschool I wouldn’t have entered my career path (software development). Same goes for accountants, office workers, administration, IT, networking, etc etc etc.
Ok, I will amend my statement, and ban computers until High School.
Me, I’d allow calculators/computers at perhaps the junior high level or at the latest, senior high. I did not really use a computer until 7th grade, then again it was a TRS-80 Model-I back then, it was 1979 after all. 1982 was a real watershed with it came to computers, I was in 10th grade and used the number of Apple ][+’s we had plus I got a TI-99/4A for Christmas that year. I think that would be a happy medium but the theory must be taught and understood first.
I agree they should be used sparingly in primary grades, more intensively in mid and high school. Most schools misuse their computer equipment as adjuncts to a typing (keyboarding) course.
While you *can* figure out most derivations by hand, it takes forever. Once you’ve gotten the concept and methodology down, it is instructive to be able to quickly show the result of a derivation and how alterations affect them.
Not quite sure of what you are deriving. Multi-digit multiplication and long division are not really derivations, they are algorithms based upon multi-digit addition and subtraction, which in turn are based upon single digit addition, subtraction, and multiplication. While you can count "gummy bears" and other manipulatives (including fingers) to arrive at correct answers, the most effective way to process this information is rote memorization of the various "math fact tables" along with the associative, commutative, distributive, and equality properties.
Having the basic "math facts" in memory allows one to build from simple (short) division with that unsatisfying "remainder", to long division and normalization of decimal points. Fractions follow naturally from division and rules for manipulation of them is again best memorized. The continuum of integers, the notion of zero as a number, negative numbers (and their rules) lead to concepts like infinity. The associative, commutative, distributive, and equality properties come to the fore again when algebra is introduced. Infinities are a good place to start discussions that introduce differential and integral calculus.
All mathematics builds from the sublimely simple to the ridiculously complicated, one step at a time, building more structure from simple facts. Digital computers are amazing tools but their internal mathematical operations are really primitive. Digital inputs are converted to base two and fed to hardware logic circuits (or software simulations) that perform addition. If subtraction is called for they perform a "ones compliment" on the subtrahend and add, multiplication is provided by multiple additions and division by multiple subtractions. The blinding speed of computers allows them to use these simplified methods to process data at prodigious speed.
Computer algorithms for performing basic math functions are not particularly useful for us poor humans who think at a much less then gigahertz clock rate. However we do get to write the instructions for our hyper speed idiots and so still serve an important function. A computer program is built from smaller logical blocks like mathematics itself, it is always wise to start with the basics when dealing with complicated subjects.
Regards,
GtG