The pedagogy of Mathematics education is fundamentally broken and we've lost generations of math users because of it.
Approaching it from this sense "two points make a line, we only have one point, so take the limit" is the method today and with all respect, it's been a terrible terrible failure...even if it is "correct".
A few kids will grok it and turn into computer scientists or physicists or mathematicians of some sort, and the other 98% will take the bare minimum to get their high school education, and if they go on to college see which degree programs require math and which don't and select the B.A. degree that doesn't.
Anecdotal, but I don't know a single person who enjoyed their K-12 math education. I know lots of people who enjoy math, and found that joy in college or later, but found the educational experience of K-12 so abysmal and torturous that they completely swore off even pursuing fields with heavy Math components. There are lots of artists, musicians, writers and historians who would probably be great Mathematicians, Physicists and Scientists were it not for the piss poor job we do indoctrinating kids into math.
It's not just that students fail to learn mathematics, it's that the pedagogical experience is so negative that they swear off ever even trying. And IMHO, a very big part of that is the (to the student) endlessly pointless jargon filled inapplicable overly rigorous and formal mess that is Math education today.
I've sat in the audience on some very heated round tables about promoting STEM education in my region and I've come away convinced that getting more people into STEM is critical to long-term economic success, but students are not only not drawn into STEM, they're actively driven away from it by the pedagogical approach of what amounts to a single class every year. Students love science classes and science labs, they love shop class and learning engineering, they love computers and everything else to do with STEM except for the K-12 Mathematics education. And that loses them, it simply drives them away from all of these great (and often lucrative) fields.
More importantly, I think fields which are not traditionally Math focused, could benefit greatly from a better general Math education. Read a social sciences academic paper and see what I mean for a quick example.
> If you try to teach a little kid the power rule right off the bat, they will both be turned off of math (because it's just pushing x around for no clear reason) and have learned absolutely nothing of the important bits of calculus.
I want to agree particularly with this, "it's just pushing x around for no clear reason". The fundamental problem with learning Mathematics in K-12 is applicability. Beyond arithmetic, students have almost zero examples of why they should bother learning anything else. Most of the population gets by just fine without anything more complex than arithmetic and figuring out percentages (and even then that's a stretch).
When students go to their parents to seek guidance, their parents also don't have any idea why they have to learn all this stuff. They certainly don't need it in their day jobs and can't provide a hint as to why it's important other than the student needs to study it for the grades.
And to be honest, even if their parents do use Mathematics in their work and can provide examples, it's likely that the student's ability to relate to that work is very limited. The reinforcement that all this time spent learning Math is pointless is much stronger than the reinforcement that it's important or useful. Getting a kid to crank through 20 or 30 algebra problems is much harder than say, a 5 paragraph essay for English class because the entire time they're doing this they're saying to themselves "why am I doing this? Math is useless! At least learning to write a little has some kind of use!"
But children will play "pointless" games for hours and hours and hours - and not even ones they're especially having fun with, just ones that hold their interest (if you've ever watched a 9 year old vent frustration at their Xbox you'll know what I mean). If we can turn Mathematics education into a kind of "game", then fill in the details and formal bits as they age, they'll at least be able to relate to it even if they don't understand the application or relevancy.
And the truth is, once you get to Calculus and get it, it's actually pretty fun and pretty easy. That's a high enough discipline for most STEM jobs and I firmly believe that every K-12 student should be able to do what we call "college level" Calculus by the 10th grade. So why not try to capture the things that make Mathematics at that level fun and easy, and I think that's the symbolic manipulation, even if it is hard to establish relevancy, and get them used to doing it from a very early age. Kids can move blocks around and stack them before they can walk, why can't the blocks be bits of equations? And why can't moving the blocks have little game-like rules they can learn?