I heartily disagree with your initial point that you can only get so far with real world examples. Heron's formula itself is not even unintuitive, and in fact, the way he proved it, requires a minimal amount of symbol pushing. Maybe my opinion is unpopular, but I believe that anything that is not applicable to the real world and does not form the immediate base of anything that is applicable to the real world has no place in highschool education.
Those of us who do math for math's sake are mathematicians. For the rest of us, math is a tool, no better or worse than any other tools we have in problem solving. There is nothing wrong with benefiting solely from the end conclusions of math, without engaging in it unless we have to. For the majority of people, this suffices and anything more has no benefit, and is promptly forgotten.
So why, then, do we insist that a high school program, tailored to the majority of people, teach people to be proficient at theorizing rather than applying? The majority are accountants ... and doctors, and lawyers, and delivery people, and repairmen, and so on. People can learn about abelian groups in the future, when they know what they want to study. Let's be realistic here. Right now, we're dealing with people not being able to do middle school material 2 years out of college. We're trying to prepare people for what they're in for. For the sake of the public good, abelian groups and math as a whole can wait.
>For the sake of the public good, abelian groups can wait.
Amazing. I can't wait to put that on a plaque. "Abelian groups can wait for the sake of public good." Holy cow! Dude, Niels Abel was 17 when he invented most of the machinery that goes by the name Group Theory. By the age of 19, he had proved quintics don't have a general solution by simply cranking up his machinery. And today we have 17 and 19 year olds who not only don't know what a group is, but don't particularly want to know. Because, like you say, they'd rather be real world accountants & repairmen, not ivory tower mathematicians. Wouldn't you rather have just 1 Abel and a whole generation of pissed off accountants than the other way around ? I would. You know, when we do eventually get the hell out of this planet and conquer other dimensions and populate new worlds, get beyond this ethereal realm so to speak, it would be purely due to the ideas of an Abel or a Gauss or a Riemann. Even an army of real-life accountants wouldn't get you off this planet - they'd be busy calculating the price of the spacecraft with their fancy spreadsheets.
I guess we have different views of what's important. There are maybe 2-3 Abel's per century. Gauss completed some of his greatest work in his early twenties, but that doesn't mean that everyone else did as well as he did. I'd love for everyone to be Gausses and Abels, but the truth is, they aren't. Neither do they want to be. Neither are they unhappy "pissed off accountants" because they aren't, like you describe. Neither are they idiots because they aren't math-savvy, like you imply.
I'd rather human development and quality of life come before space travel. If there are junior Abels among us, they are going to stand out regardless of whether you teach them subject X in highschool or not. Furthermore, they are going to pursue a specialized education (college) in said subject anyway. I don't think not recognizing geniuses is a problem we have since geniuses tend to be fairly resourceful. Thus, we can focus on improving the standard of education for everyone else. I'm not suggesting a ban on the study of mathematics altogether.
There is no causation link as you imagine. No amount of increased math in highschool is going to breed geniuses like Abels. Increased general skills like logic and problem solving is going to result in higher ability and better education for almost everyone.
Those of us who do math for math's sake are mathematicians. For the rest of us, math is a tool, no better or worse than any other tools we have in problem solving. There is nothing wrong with benefiting solely from the end conclusions of math, without engaging in it unless we have to. For the majority of people, this suffices and anything more has no benefit, and is promptly forgotten.
So why, then, do we insist that a high school program, tailored to the majority of people, teach people to be proficient at theorizing rather than applying? The majority are accountants ... and doctors, and lawyers, and delivery people, and repairmen, and so on. People can learn about abelian groups in the future, when they know what they want to study. Let's be realistic here. Right now, we're dealing with people not being able to do middle school material 2 years out of college. We're trying to prepare people for what they're in for. For the sake of the public good, abelian groups and math as a whole can wait.