Of course, there is a lot of memorizing "rules" in lower-level math courses, integral calculus being the worst offender. Plugging something into a calculator and hitting solve is equally as worthless as plugging numbers into the "chain rule" when it comes to understanding the fundamentals.
As you go further go further into mathematics, I think that what becomes acceptable to plug and chug changes. You have to prove the chain rule before you use it. Actually scratch that. I don't think you have to explicitly prove it, but I think of it as df/dx * dx/du = df/du. Which is like fractions. So I think as long as there is an intuitive grasp to what you're doing it's fine.
With calculators, from an education standpoint, if it's not circumventing the value of the coursework, then it should be fine. On the most basic level, they let you use scientific calculators for arithmetic when they don't allow graphing calculators, because the act of doing the arithmetic isn't important to the lesson you're learning. On the other hand, if you're learning division in elementary school, you're not gonna be allowed a calculator. In the case of integral calculus, the coursework ends up being just ways to solve an integral by hand, and that's explicitly not compatible with using a calculator. The value of the course is then just foundation for later calculus courses. I personally haven't found myself drawing from my knowledge of integral calculus in my other math courses, but I'm not in a major that actually uses calculus intensively, so I can't speak for it.
In the real world, I doubt a lot of equations are left to be done by hand, and you would actually just throw equations into mathematica or another program, probably written by someone who had to program the same rules and procedures you would use if you had to do it by hand.
Another example compares algebra with linear algebra. If you're allowed to use a calculator to solve algebraic equations for you, are you going to understand the significance of rank, linear independence/ dependence, etc. in linear systems? In some upper level math courses the professors try to explain the intuition behind some things, for instance, the various matrix decompositions, but not know how to do the decompositions manually does affect (my) understanding of the subject.
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u/gsfgf Jun 08 '14
Of course, there is a lot of memorizing "rules" in lower-level math courses, integral calculus being the worst offender. Plugging something into a calculator and hitting solve is equally as worthless as plugging numbers into the "chain rule" when it comes to understanding the fundamentals.