I don't recall the integer based sqrt algorithm, akin to standard long division in structure, featuring in any of my K-12 text books for teacher led discussions. Here in this archive, I squandered a piece of my adulthood learning it late and rewriting it in Python so I could make it go faster. I don't regret the exercise. I'd be happy to do it again here sometime.
Going back to my standard line: I think the college math buffs will have an easier time recruiting from my geek corps of gnu math graduates. They'll not be soured on the subject, plus appreciate the historical significance and ongoing importance of such as the square root algorithm, Euclid's for the GCD, completing the square, along with more recent algorithms for finding a convex hull, linear programming or whatever.
But it'd also be clearer to them, as it's already apparent to us, that this version of K-12 quite appropriately has no "boot camp" for wannabe mathematicians, nor a "weeding out" process vis-a-vis this particular department, as in "controlled by its inner sanctum minions."
That'd be too narrowed and focused in a time of life when overview and "how things work" should be of more central concern. Go through those gates later if you like, but don't expect your public school system to kow-tow to some top ranking math general in some army of mathematicians. Let your post K-12 math teachers play by those rules, after students have had ample opportunity to survey the fields and make some choices. Some will prefer computer science, another branch of engineering.
K-12 wasn't so weighted towards pure math teaching that these other doors weren't even opened.
But what if that's your goal, as a kid, to become a pro mathematician, an inner circle math giant? Well, by all means supplement then. The K-12 I'm describing won't hurt your chances I don't think, will give you a boost. But don't expect it to warp into a completely different shape. Take responsibility for your own self-education, like a real mathematician (it's never too early to cultivate the right habits of mind).
And by K-12 I mean some archetypal public school that doesn't really exist except in the governmental blueprints, like abstract classes in Java.
But anyway, it's numeracy, a cultural mix of stuff, that we teach. I'm sympathetic to Everyday Math using a lot of pages to teach geograpy, not sympathetic to the between the covers promo of calculators. Schools sytems shouldn't have to pay publishers for the documentation the calculator companies should be providing online.
But geography, that's useful, because the planet is spherical and trigonometry pertains. And because the logic of latitude and longitude has sophisticated mapping relationships to various whole world projections that're flat instead of round. So yeah, we're doing a lot with geography in numeracy training (XML, GIS/GPS... Google Earth). And now let's include astronomy as an extension of geography (Celestia... Google Moon).
A mathematician comes along, sees all this "concrete content" encroaching on what in his mind is supposed to be closer to a pure Platonic philosophy, and says "wait a goddamn minute, this isn't *math*!" -- by which he means something more like Adrian in this archive: a sequence of definitions, axioms, proofs, more definitions, more proofs, tweak an axiom, alternative paradigm and so on. A few pictures, a joke or two.
My reply to the mathematician above, with respect to our K-12 charter schools for geek kids on the Pacific Rim (think of gypsies, open camp fire along the road someplace, lots of LCD screens), is "you're right, this isn't math!". Then I might flip up my laptop lid and show the ~M! emblem, glinting in the firelight, a trademark of the made-for-TV series I'm working on.