Hm, you're talking about teaching symbolic manipulation using calculus, but for me the real ideas of calculus are integration and derivation. I think those ideas are very valuable, and symbolic manipulation is just one way to solve those problems.

 Teaching integration with various shapes and boxes of ping balls is really fun. Then use smaller balls to get a better estimate. Physical, visceral and chaotic with balls bouncing everywhere. Even counting a hundred-ish balls is hard and methods have to be devised (good way to intro number bases).Exercise can also be done in 2d with colored squares or circles. Which is a nice segue into guestimation. Cover the entire curve to be measured with a square, value can't be large that as the square totally covers it, ask them to find the lower bounds.Then get them to make estimates with other things they have around (bananas, shoes, etc), someone will come up with rectangles. Introduce the trapezoidal rule!Then we are off into history with Newton and Leibnitz and that shit is good drama enjoyable by any age.None of what I have presented requires counting past 20 (which really most of us can't do anyway).
 >but for me the real ideas of calculus are integration and derivationyou mean the ideas of a dual space of a linear space and of tangent bundle to a manifold? :)
 Why not? :) All these things are really simple in their essence. They're abstract but an average 14 year old in modern civilisation already has the cognitive machinery necessary to grasp such abstractions. In fact, I think at that age, when they discover abstraction and formal reasoning and what they can do with it (there's a reason teenagers are so full of all kinds of ideas, they're starting to explore the world in full generality), it would be extremely stimulating food for their minds.
 Can you provide some examples? To me, the basic mechanics I remember (for simple equations like x^3 d/dx) were almost entirely "solvable" by moving bits of the equation around...1. move the 3 in front of the x2. where the 3 was, subtract 1 from it and put that back where the 3 wasbut if you think of it as symbols you get more like1. move the exponent symbol in front of the x symbol2. subtract 1 from that exponent and put it back where the original exponent symbol wasFor me at least, bridging the gap from the specifics of the first example to the more generalized second approach meant that when I was given something like3x^(y^4) d/dx my mind was absolutely blown because I didn't know if there was some rule I needed to know if the exponent had an exponent or something.Going from the years of the arithmetic approach to really groking the symbolic manipulation approach was really hard for me, and looking at lots of young kids suffering through algebra it was the same.Learning how to recognize the symbols through the specificity of the numbers was something I don't think I really got at all until calculus and it's because the mechanics of the rules of simple derivatives and integration really force you to recognize the underlying symbols...more than algebra did (which for me just seems like a really complicated arithmetic rather than what's actually a pretty simple symbol manipulation exercise).It's odd, because like most technically minded kids, I was already great working with symbols and breaking things down and building them up. I could build huge structures out of various small assemblies with legos, write simple software, crank out papers for English class that would guarantee an A and so on, but understanding that the teachers no longer wanted me to be a human tabulation device but to be a manipulator was something I never really cottoned on to.I know from spending a little time with nieces and nephews younger then 10 that you can teach basic derivatives in about an hour or two and have them fairly reliably doing simple work like the above in an afternoon if you don't worry them too much about simplification of the result or all the arithmetic.My gut says that if you start adding more rules, like what do you do when you have something like (x^3)/(x^2) and how exponents should be subtracted, and with a little care in the examples you show them of this rule in action, you end up showing them all of algebra, fractions, exponents and later trig, logs, etc. while building up and hanging all of these concepts off of calculus.TBH I don't know enough about early childhood education to know if this should be used instead of basic arithmetic or not, but I bet if you use shapes and colors instead of letters and numbers and start to teach basic rules, you can just slowly introduce numeracy later anyways when their brains are more developed.
 No offense, but what you're proposing would be a huge step backwards. All of those examples you've given are extremely special cases of a general concept, where the concept is relatively easy to understand (two points make a line, we only have one point, so take the limit) although the implementation is complex (requires either understanding algebra or knowing lots of rules.)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.
 > where the concept is relatively easy to understand (two points make a line, we only have one point, so take the limit) although the implementation is complex (requires either understanding algebra or knowing lots of rules.)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?
 >Read a social sciences academic paper and see what I mean for a quick example.I have actually done some graduate-level work in sociology and history, and the papers and books I read were mostly examples of very good statistical work and well-thought-out process analysis. You can put your STEM-master-race badge away.>Beyond arithmetic, students have almost zero examples of why they should bother learning anything else.When I say "for no reason," I don't mean "for no day-to-day practical reason." Playing with abstract concepts is and should be its own reward; that was the whole point of TFA. Mechanically memorizing how to take the derivatives of polynomials is neither a fun abstract concept nor a boring-but-necessary practical skill.>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."Gamification" as a cynical ploy to get kids to sit still long enough memorize their times tables may or may not work. But even if it does, it's only gotten them to play the game long enough to pass them to the next level; it has deliberately shifted their interest away from the joy of learning for its own sake. That is not what the article is about, and it's not helpful in the long run.>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".No... no, it isn't. The approach today, for the majority of students, is to learn the bare basics so that you can plug them into an equation and find out what the marginal cost of widgets will be next year given a certain set of equations. And in any case, it comes so late that kids have been taught that "math" is something that actually is boring and useless.