Finding some connection between trigonometry and angles of rubber-powered trebuchets is incidental and quite frankly, this approach isn't going to take you very far. One simply can't find real world mappings to atleast 99% of what is properly called math. Even if you'd like to strictly operate within the confines of high school geometry , it is a study of points, lines, planes, angle relations, norms & distances motivated by axioms & theorems that follow by symbol pushing. For example, it so happens that if you push the right symbols, you'll eventually work out that the area of a triangle is related to its sides ( http://en.wikipedia.org/wiki/Herons_formula ). But this is a consequence of your definition of area and triangle and side and so forth. The relationship falls out of those definitions. Its not about constructing a real wooden triangle out of real sticks & then measuring its area and then the lengths of its sticks and saying - hey, I told you so! Those things ie. the real wood and the real sticks,...reality - is just a sideshow. Real world problems are at best a distraction. If you were to focus only on these sort of real-world mappings, you'd end up as an accountant - because that's were 1 would map to $1, 2to $2 and add would mean 1 + 2 = $3 and so forth. That's what corresponds to reality and the real world. That isn't math. Math is where you say 1 + 2 = 0 because the abelian group G3 with three elements {0,1,2} can satisfy the addition rule and have three distinct elements only if 1 + 2 was 0. For if 1 + 2 was 1, then that would mean 2 was equal to zero by cancelation laws. Similarly if 1+2 was 2 then 1 would equal 0. Since you do have 3 elements, it stands to reason that 1+2 must be 0, because it can't be anything else in that system. That's math.
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.
You sound like someone from the Konishi polity as described in Diaspora. I don't agree with you. The brain uses an associative learning algorithm and learning difficulty is proportional to how easily a concept can be integrated with what is already known. And it's an iterative process, so new knowledge can be built on old knowledge that was once itself abstract.
So you can start with concrete problems and generalize from there. No one starts out learning about unitary matrices, hilbert spaces, and natural transformations. By the time they get there they can be taught in terms of more concrete things like rotations in 3 dimensions, euclidean spaces and parametric polymorphism (if they are programmers). That is why math is hard when you first start. It's a highly compressed subject where every word is more an index than it is a full concept. So you have to maintain a cascade or hierarchical graph of intuitions.
>you can start with concrete problems and generalize from there
You could. But that isn't the only way. You should really read up on the way math is( was ?) taught in gottingen & eastern europe & romania & russia. Speaking of which, I learnt most of the machinery behind matrices in India in high school, with no reference to where it came from or what its applications were. I could compute the determinant of square matrices, compute adjoints, inverses, multiply two matrices....all of this without knowing wtf a matrix was or what its real life use might be. This is true of most high school students in India, simply because of the way the curriculum is structured. Its only when I took at a graduate level computer graphics course in my late 20s did I see that linear transforms were actually matrices, so you could do rotations and translations of vectors by matrix multiplication. So real life applications don't have to precede abstract concepts - you can make a hell a lot of progress by approaching stuff the other way around.
> I could compute the determinant of square matrices,
I learned to do this too---and then promptly forgot---several times over until someone finally clued me in to the fact that the determinant is the scale factor of the linear transformation associated with the matrix. Then everything clicked.
It's not just that I couldn't remember how to define/calculate the determinant, I also couldn't apply it usefully in my (proof-based) linear algebra class because I didn't have any intuition for it or any way to connect it to other concepts.
