' You get a whole lot of concepts that most people will simply never use again -- polynomials, congruent triangles, logarithms, the quadratic formula, etc.'

Very true, but if you have a class of young kids how do you know at that age the ones who will eventually need those concepts from those who won't? Obviously, some will never need to know even relatively simple concepts such as, say, having to know the focal point of a parabola in Cartesian coordinates whereas others may perhaps become electrical engineers who need to know Maxwell's [partial differential] equations—and even then some will have to go further and need to know the relativistic incarnations of those equations (which is a necessary skill to solve some practical engineering problems such as determining the precise efficiency power alternators).

Whilst it's possible to plug in values into existing equations (and algorithms) and achieve an outcome (as one looks up sin and log tables without needing to understand how the tables were created), I find it difficult if not inconceivable for engineers or scientists who, say, work with fields and dynamic flux changes etc. to be efficient at their work unless they have some ability to mentally picture vector fields, etc. Clearly, learning how to manipulate functions using calculus techniques provides the prerequisite training necessary to develop these skills in any meaningful way.

If you do not inculcate the basic ideas and concepts that underpin calculus sufficiently early on then we are likely to lose people from those skilled professions. At what stage in one's education we introduce these concepts is, no doubt, subject to much debate but it seems to me the earlier it is the better (if for the very reason that a grasp or notion of the power of mathematics is often integral to the formation of one's idea of entering a profession that requires mathematics).

You're right, many will never use many of the concepts you've mentioned again, but it makes sense to me to introduce kids to the notion that there's more to mathematics than just counting, multiplying and dividing integers in ways where all numbers involved must be directly accounted for. Even quite young kids can understand the concept of converging infinite series and when given ideas such as Zeno's paradox they quickly understand the paradox and can see that there's a sneaky way to solve the problem (even if they yet haven't mastered the technique). In fact, they even get considerable enjoyment out of the fact they can 'bypass' the need to count to infinity by using a form of mathematical 'cheating'. (For instance, the frog example used in my earlier post was taught to me in primary school and it was used to explain what calculus was about. We were told that when we got to high school we'd learn that if we knew the size of the frog's feet then with calculus we'd be able to calculate how many hops he'd have to make to get to the pool's edge.)

Even if we don't teach kids advanced calculus concepts, we still need to teach them the basics such as the concept of limits and Simpson's rule, etc. as these are essential mathematical concepts whose notions extend into all manners of real-world instances. I fully agree with you on the matter of pragmatic training. From my experience, a great deal of the mathematics I was taught at high school was done without any reference to practical real-world applications and I believe this is the wrong approach for two reasons: it's easier to grasp the concept if you can see how it applies to a practical application and knowing how the mathematics applies to the application makes it all the more interesting and thus more likely learned thoroughly.

(The trouble is mathematics curricular are usually set by mathematicians or those who have an intrinsic love of mathematics and who think it's unnecessary to apply mathematics to solving real-world problems to appreciate its beauty. Unfortunately, the percentage of people within our communities who think this way is comparatively small and thus I'd venture that it's too small for them to be sufficiently representative so as to be able to determine the mathematical requirements for everyone. I reckon we need to break this nexus but I'll admit there seems no easy way to achieve this.)

I recall a good example of this problem was when we were introduced to the concept of parametric equations. To me, introducing the parameter was a really nifty trick but some kids just couldn't get it and found it all very boring—nor did the teacher bother explain its uses other than we were required to know it so as to pass our exams. Likely, this could have been eliminated from the syllabus, as it seems there's no simple or practical real-world application to give students at this point. It wasn't until much later I found out about other cleaver mathematical tricks—albeit ones that are in ways very different to my parametric equation example—such as those used by physicists in quantum field theory to resolve troubling infinities [renormalization]. Mathematics sometimes doesn't always cooperate in resolving practical problems (albeit complex ones) unless we resort to using such tricks.

Even being aware of the need to apply 'tricks' such as renormalization is way beyond the scope of both high school mathematics and science, thus it begs the question at what stage should notions such as 'complex manipulations' be introduced into mathematics. If we introduce them too early we risk alienating students through both their complexity and irrelevance, do so too late and the shock of such seemingly irregular and unfamiliar techniques will likely overload the student at a time when he/she should already know such techniques off pat.