> Playing with abstract concepts is and should be its own reward

I'm sorry, but you're just simply wrong on all points. Promoting the status quo in math education, as you are doing, has been, is, and will continue to be a failure that drives kids away from learning. There are now decades of evidence of the failures of k-12 education to address this need and I find it unbelievable that you haven't gotten the picture yet.

I'm not saying that what I'm proposing is correct, but continuing the very poor pedagogical approach that you support is not going to solve the educational failures that we're experiencing today. What we need are fundamentally new approaches to Math education. You are not providing any insight into what those approaches should be.

I'm sorry, but in a discussion to fix and change what is obviously utterly broken in k-12 maths education, suggesting to just continue the course is not a helpful contribution and is simply part of perpetuating the problem.

This has been recognized for so long, that it has finally percolated out of educational establishment, which has failed to address the problem with undereducated and unqualified teachers, student motivation, repeated failures in curriculum development (Common Core is simply the latest joke of a curriculum), and has reached levels as high as the White House for targeting. You have to know that the current approaches are failing if the President has to get involved.

From the "Report to the President...K-12 education in STEM"
http://www.whitehouse.gov/sites/default/files/microsites/ost...

"Schools often lack teachers who know how to teach science and mathematics effectively, and who know and love their subject well enough to inspire their students. Teachers lack adequate support, including appropriate professional development as well as interesting and intriguing curricula."

"As a result, too many American students conclude early in their education that STEM subjects are boring, too difficult, or unwelcoming, leaving them ill-prepared to meet the challenges that will face their generation, their country, and the world."

"Put together, this body of evidence suggests that grade-school children do not think as simplistically about STEM subjects as conventional curricula assume. They are capable of grasping both concrete examples and abstract concepts at remarkably early ages. Conventional approaches to teaching science and math have sometimes been shaped by misconceptions about what children cannot learn rather than focusing
on students’ innate curiosity, reasoning skills, and intimate observations of the natural world."

http://books.google.com/books?id=7v2gRGuxw4sC&pg=PA253&lpg=P...

"The first principle of How People Learn emphasizes both the need to build on existing knowledge and the need to engage students' preconceptions -- particularly when they interfere with learning. In mathematics, certain preconceptions that are often fostered early on in school settings are in fact counterproductive. Students who believe them can easily conclude that the study of mathematics is 'not for them' and should be avoided if at all possible."

I differ from this report in that they continue to promote a bottom-up approach to maths education. I think math should be taught from a top-down approach, like nearly every other discipline. You don't learn to bake a cake by first spending 10 years learning about chemistry, agriculture, nutrition, animal husbandry, distillation etc. You say "I want to bake a cake" and you start with a simple cake recipe. Then the next time you say "I want to bake a different cake" and you use a more complicated recipe. And so on and so forth until you don't need a recipe and are putting together your own cakes from scratch.

The "cake" I'm proposing is calculus...and I believe, from experience teaching basic calculus to kids under 10, that this is realizable and beneficial.

The end result is that I don't expect kids to come out of K-12 baking cakes that look like this http://www.askmamaz.com/wp-content/uploads/2013/01/Most-Beau...

when the average STEM work-a-day mathematics really requires them to do this.

http://www.recipegirl.com/wp-content/uploads/2013/06/Red-Whi...

Right now, your approach is producing students who don't even bother with cake baking, and when pressed into service produce this http://fc05.deviantart.net/fs70/f/2012/070/1/b/ugliest_cake_...

This needs to change.