"But," you say, "Jon, don't you teach math?"
Why yes, yes I do. I like math; apart from some issues with algebra in high school I am and have been fairly good at it; I majored in math at Hillsdale; now I'm teaching it. I am not trying to suggest the the study of number is useless or otherwise overrated. I simply suspect that it is overcomplicated in the modern school.
I would say there are three broad uses for mathematics. I am going to call these the ordinary, technical, and artistic uses.
The ordinary math is practical. This is the basic math that enables you to estimate a grocery bill or calculate the length of a trip. At its most essential, this math is little more than the knowledge of facts like the infamous multiplication table, a grasp of simple concepts like unit conversion and averages, and an acquaintance with the concepts of probability.
Technical math is also practical but of a different order. For one thing, it includes most math, from the geometry used in surveying to algebraic and calculus models to set theory and probability to frankly I only have the vaguest idea where most concepts are used precisely but you try doing anything tech-y and you run into math. But the key difference is that the use of this mathematics is defined by the necessities of a profession: unless I wildly miss my guess, a tailor won't need the same math as a nuclear engineer.
Artistic math is math as its own thing – we might also call it pure math. Whatever we do here may have applications, but that is not the primary purpose. I think it would also include math puzzles and games like sudoku.
The situation, roughly speaking, with (at least American) math eduction as I have experienced it can be explicated as follows. The goal is to teach technical math: in order to fit in a "complete" background what is actually be taught is only artistic or pure math: but so much is covered that even this is taught too much in the rote style necessary for practical math.
Do we expect too much? Perhaps. In other ways, we expect too little. I use a widely-standard textbook series: from Pre-Algebra to Algebra 2 is supposedly three years' worth of study, but there's so much overlap that it should be two years... if that. On the other hand, many of the basic concepts are cluttered up with fancy formulas with little differentiating the necessary from the curious.
My own theories of education, such as they are, draw heavily on a dictum ascribed to Churchill. He is supposed to have said the important thing was for students to learn good English, after which he would allow the clever ones to learn Latin as a reward and Greek as a treat. We can of course quarrel about the bare minimum necessary – much of the new classical movement, for instance, considers Latin essential especially considering its influences on English grammar and vocabulary. The point, however, is that I am not convinced we serve students best by insisting on amassing unnecessary knowledge. Let me illustrate: for my current work, I don't even need to know Calculus. Yes, it helps to have a better conception of "the whole"; but the quadratic equation is unaffected by the fundamental theorem – and I teach the stuff. But the impression I get is that we encourage more and more students to take Calc – either by state standards or private school snobbery – while actual basic match scores continue to fall.
My current conclusion is that we demand too much in terms of quantity (of information) and not enough in terms of quality (of knowledge). While I have some thoughts on how to improve this, they're relatively incoherent, and my experience is limited, so for now I'm just going to throw the problem out there to remind me to revisit the subject later – I've been meaning to write on this for months now.