Rant about math education

For leading a pretty fortunate life, I still seem to find an awful lot to complain about :-)

"Logic merely sanctions the conquests of the intuition." --Jacques Hadamard (quoted in this article about math education).

I both agree and disagree with the article, but agree 100% with Hadamard.

High school math class focuses on applying tried-and-true processes to pure abstractions: long division, solving equations, rules of differentiation and integration.  If a kid struggles with these tedious, mindless, and mechanical operations, they are pronounced to be "bad at math".

But, who in the world (short of a few number theorists and mathematical software and hardware developers) really cares about knowing how to do these?   Almost every kid now has a cell phone that possesses enough processing power to do thousands of these operations accurately and within milliseconds.  

In many engineering and scientific domains, it is indeed a little important to know the mechanics of integration, but it is much more important to know when to use integration, what to provide as input, and how to interpret the results.  I felt my college-level math classes did a good job of addressing these questions.

College math also taught me the following: estimating the error in the measurement of initial values; how to determine the sensitivity of a calculation's result from error in initial values; how to determine sensitivity introduced by a particular algorithm that is used to implement the calculation, because of the finite-precision of the operations; and how to use all these factors to determine the accuracy of the final result.
  
What it didn't teach, and I wish it had, is:  how to derive a model that describes a system; how to concretely formulate a problem and understand what I am trying to solve; how to accurately gauge what assumptions I'm making about a system based on my model; and how to discard terms in my model and simplify a calculation, without significantly comprising the accuracy of the solution I'm seeking.   For that matter, how do I know what constitutes a significant compromise of accuracy?