Do you have any resource that you can point me to that speaks about the relationship of limiting constructions to irrational numbers? (Not being the usual /. snark, an honest question)

I guess that I ask the question because there are perfectly boring shapes that give rise to irrational numbers, like a the hypotenuse of a right triangle with two sides of length one.

There are also a lot of rational limits that converge to rational numbers.

I have just never heard of any explicit connection.

True but that 17 year old had a job that didn't require the same type of prerequisite education. Not all talents translate into the type of "book" intelligence that we expect of people in modern society. They had more "natural" talents at that age that translate well to the jobs of the day (like agriculture) I think that we forgot how "unnatural" abstract though is for most people.

Of course I think that we, mathematics teachers, often doom our students by having low expectations. I just am suspicious of what would occur when the beautiful curriculum hits the class. Thinking about material seems to run so contrary to their understanding of what learning and school is that when presented with this "new" approach many violently reject it. I know we need to find a balance that doesn't sacrifice the education of the "motivated" students (I teach at a community college where I have many very intelligent kids who were just so put off by the HS curriculum that they actually failed) but also doesn't just forget about the, I hesitate to use the word, bottom 3/4 of the class. How does one do this? I do not know, and it makes my head hurt. (which may be why I will leave this environment and get my PHD, much easier)

I applaud your general point, that people are generally lazy and prefer dogma to critical thought, but I have to disagree with your prescription for US mathematics education. (at least in practice) Having students attempt to have been exposed to Linear Algebra, DiffEq, Multivariate Calculus, etc... before the end of high school will have the opposite effect of the one that you desire. I have already seen where many of the better students have been pushed through single variable calculus in high school before they have really deeply learned the prerequisite material. Most of the students coming out of this system will have a MORE shallow and mechanical (dogmatic) understanding not less so. I think that a slower and deeper introduction (with a good foundation in mathematical logic) to the more fundamental portions of mathematics will result is a student that is more prepared to appreciate and understand the scientific material that they are being exposed to. *note that all of this requires children to be more mature then most prove to be, how do we deal with that? *