Not being a teacher, but having many years' experience in various workplaces, I find this argument interesting.
There are now tools that allow much of the mechanical aspects of math to be done with little human effort. No one would have asked me to study science without a slide rule. And no one in the business world tries to do a lot of arithmetic without an adding machine, a calculator, or a spreadsheet. It is fruitless to insist that students eschew the labor saving devices now available, labor saving devices that they will be expected to utilize effectively in the workplace.
Assuming anyone considered my opinions on pedagogy worth considering, I would take advantage of the tools now available to spend much more teaching time on concepts and word problems, which require figuring out what concepts to apply. What is worrisome, however, about tools, formulas, rules, etc. is that, without understanding, it is very easy to apply the wrong one to a specific problem. One of the things done well by mechanical problems such as what is the derivative of this function is making concepts concrete and thereby solidifying understanding. So I would still devote time (just less time) to solving mechanical problems without utilizing black boxes. But such mechanics are not the essence of math. Do any of us here think that Jason, no matter how diligent he is about learning rules and how careful he is about neat presentation, will ever be able to apply calculus?
