(3) Thirdly, the linear push to calculus is a turn off to many students. The style of teaching most HS calculus teachers are forced into teaches students that advanced math is not about thinking but memorization.
Instead of answering the question directly, I'll talk about this a little, and some of my own high school experiences. I graduated from high school in 2008, so the things I say are current & relevant...at least around here (suburbia of St Paul, MN, USA). There was one time when underlying mathematics was emphasized: intuition, concepts, etc., and that was in maybe 2nd grade of elementary school. One of the most memorable things was this. We had magnets with our names on them, and in the morning, we always put our magnet on the currect side of the side of a metal file cabinet. On the left, we have "Hot Lunch" (school-provided lunch), on the right, "Cold Lunch" (you brought your own lunch). This was so the school could estimate how much to make during the day, I guess. Anyway, one day, there were nine magnets on the "cold lunch" side. The teacher then asked me this question in the morning: "Robert, I have nine magnets for cold lunch. Can I arrange the magnets into a square shape?" Now, we haven't studied anything like this before, so this wasn't a drill question at all. Since I was being "singled out" in a way, I was nervous and didn't know what to say. But the teacher encouraged the class to think about it. We learned a little about multiplication, and I said a bit later "well if it is a square, then the number of lines [rows] should be the same as [the number of] tall lines [columns]" and then "we take those and times them and the number we get is a number we can make squares with". My logic was all mashed up and confusing when I said it, as you can imagine, but we all learned this idea of "square numbers", when knowing only a little bit about what multiplication is, and such. It was a time where we *really* learned, and we didn't memorize anything. Skip to third grade, that's when memory+drill started. Times tables, timed tests, all that stuff. 3: addition, subtraction, multiplication, division, timed tests, long division 4: weights and measures, time, "word problems", area, volume, basic geometry 5: more basic geometry, number properties, calculations/calculators, basic algebra problems 6: more of the same, graphs 7: more of the same, basic algebra, deciphering word problems, "exploratory" stuff 8: Algebra I: solving stuff, quadratics, modeling, graphs At this point, I personally went the "advanced route". Usually you'd only have a year of calculus if you went the basic route. Even then, calculus was optional and you could replace with something else. 9: Algebra 2 + Geometry: more algebra, geometry, two-column proofs 10: Precalculus: trigonometry, circles, triangle solving, sinusoids, vectors, matrices, graph transformations (XY plots, not "edge/vertex" graphs) 11: Calculus 1: limits, differentiation, methods of diff, applications, riemann sums, integration, basic methods, fundamental theorem 12: Calculus 2: methods of integration, related rates, differential equations That's a brief outline of how things went. Now, I want to emphasize *how* it was taught. 3: drill drill drill, you better know 1 digit multiplication and 2-digit-1-digit division 4: This was memorization, but these were practical things you needed to memorize, like grams and kilograms and hectometers. There was terminology like "parallel" and "perpendicular" 5: Memorize area formulas, memorize divisibility, learn how to factorize integers, learn how to do multi-digit multiplication 6: Memorize how to do more problems, memorize how to properly make bar graphs, histograms, box-and-whisker plots, etc. 7: This was not very memorize-oriented. This was more concept oriented. Good teacher. 8: Algebra drill. Factoring, FOIL, expansion, formulas, all that. 9: Memorize how to do two column proofs + the names of logical arguments ("modus ponens", modus tollens (spelling?), contrapositive), learn about reflection, dilation, etc, memorize how angles relate 10: Memorize trig identities, trig values 11: Memorize limit laws, differentiation laws, integration rules 12: Memorize integration rules, differential equation junk Anyway, the point is, above all, everything is so drill based. Everything is so memory based. If you don't remember, you're penalized (exams). This is an obvious problem that's been pointed out a lot, so I won't dive into that issue. There is another issue I had though. I was largely self-taught. I bought my own books and learned all sorts of stuff on my own. The biggest motivating book for me was "Mathematics: From The Birth of Numbers" by Jan Gullberg. This was and is a wonderful compendium of mathematics, written by someone who also taught himself. You can read my review of it here: http://www.amazon.com/review/R2QT66JA1XGJWG Anyway, being self-taught means I could go ahead of the game. There were two huge roadblocks though from letting me get ahead formally. The first one was that to get ahead "formally" (skip grade levels, etc), I had to know what they were teaching. This sounds innocent and fine, but in reality, it is ridiculous. I could demonstrate that I knew how to do trigonometry, but since I couldn't make a box-and-whisker plot proper, that means I did not know enough to move on. But multiply this by 1000. There were lots of these little random things with no importance to math or anything at all, but were required to know -- that most forget a week after anyway. So what if I can't do a compass-straightedge construction of something when I'm in 9th grade. I know the core curriculum, why can't I move on? That was a small one, but it's largely forgotten about and shadowed by an even larger one. I could competently do calculus in 9th grade. I took the AP Calculus BC exam then, and scored a '3'. Not a wonderful score, but for being self-taught entirely, I thought it was grand. A lot of colleges would probably accept that. The better universities, probably not. But I knew what they tested for and I could get a better grade next year anyway. And the stuff I suspect I missed were mundane things anyway. Anyway, so I "basically" have college credit for one semester (one year?) of university calculus (two years of high school), which is single-variable calculus. I also got the top score on the AMC12 (American Mathematics Competition 12) in my school, when I was in 9th grade. I asked the school if, in 9th grade, I could perhaps go to the calculus classes at least, or maybe even go to the university for math classes. They said "no". I argued with them on and on (by 'I', I mean 'we', my father and I), and they granted me to take a test on algebra and geometry. I took it and was pretty sure I did great, but they said I failed it, but refused to show me either the test or the score. A year passed, 10th grade trig. I looked into alternatives, bla bla, asked again if I could skip, nothing came about. In this time, I taught myself some interesting things like Fourier series and began to acquaint myself with some programming stuff. Another year passed, 11th grade calculus. One day in class, I had *my* multivariable calculus and vector algebra book with me. It was just off to the side and the teacher saw it, and he said "wow, why do you have this?" I told him why and told him how I'd already taken the AP tests in 9th grade -- the very tests that the calculus class I was in was preparing us for. He said "well what are you doing here?!" and he suggested I stay after class. I stayed after and he brought me into the other room with the other math teacher, "Mr Butler", who is essentially the head of the high school's math dept. The teacher said "did you know Robert is studying multivariable calculus on his own, and [...]?" The teacher seemed genuinely excited for me; I was pretty happy about it! But then Mr Butler just kind of mumbled saying "yeah ok". My teacher's attitude changed so sharply at this point. My teacher said "we'll look into something Robert, thanks". Anyway, time passed and nothing came of it. I asked about it, and he said he will give me a test. Some time later, I took the test (which covered one year of high school calculus), and he said I passed with a remarkable score. He said I'd have to take another one though. The other teacher, Mr Butler, is the one who administered the second test, which was on "Calculus II" (high school year 2 calculus). I took it and after maybe 2 weeks, I asked about it, and he said I failed, and wasn't able to do anything. It's now one semester through my third year of high school, and nothing has happened. I'm still in Calculus I at this point. After fighting the school a good amount, they finally gave in, and said I could take a university class if I could find one. They didn't help at all with that, so I decided to just take an all-inclusive "Independent & Distance Learning" course at the University of Minnesota, for Multivariable Calculus. After getting that straightened out, the school year is almost over. Maybe 3/4 of the way through. I now go to the library instead of calculus class. I asked the administration if I got credit for calculus 1 and 2, and they said "no." I thought I had "skipped"/got credit for them, but they said no. This caused a big stir. What's the point if skipping if I won't get the necessary *high school* credit to graduate? Toward the end of the year, I finished the entire Calc 3 curriculum in one week. Two days to complete all ten homework submissions (typed up), and I scheduled three consecutive days to complete the two midterms and final exams. I finished that course with 98.9%. During my last year of high school, there seemed to be no opportunities left. There were no math classes to take without the school's help, etc etc. It took the entire year, with a threat of a legal battle (lawyers got involved and whatever), and they gave me credit for calculus 1 and 2, and credit for the four years of French which I completed in one year, and four years of Spanish which I completed in two years. All in all, the whole thing was hell. There was so much time wasted, so much money wasted. It personally affected me. I did poorly in some of my other classes just because of its affect on my mood (I did okay on exams, but in high school, homework accounts for at least 80% of the grade). I barely got through everything. In fact, I had to go to summer school twice because I failed two classes. While it was me who failed some of the classes, I don't carry all of the blame. It was, without a doubt, very stressful and very hard, fighting with the administration during the whole time, trying to get things straightened out and comfortable. In the process, I also moved to a new house, and there were some other family/personal issues to deal with. Applying to the University of Minnesota and some other universities, three or so years ago, was not a success. I was rejected from all of the universities to which I applied, as a result of poor marks in high school, I suppose. My only option right now seems to be (1) go to a community college, and wade through their system, and try to transfer, with only some of the credits transferring to a bigger university, (2) try to figure out a shortcut in the system, and get accepted to graduate school without an undergrad, (3) don't go to college. #1 is the one I dread most. I don't want to go through high school again (which is essentially what a CC is like). And there's no way I could push myself at a CC with advanced courses. I want to learn, not idle. #3 is a very tough choice. I have been applying for jobs for a long time, and have not been accepted to anything. I've done freelance stuff, but it only brings in a very small fraction of what is necessary to live. You really need to strike gold to get off with #3 it seems, these days. #2 seems like a very difficult, but *possible* outcome. I've been auditing graduate math classes at the University of Minnesota for a bit, and my hope is I'll get a connection. Or maybe from somewhere online. I also am trying to get stuff done with my book, and trying to develop publishable material (that isn't "salami science"/smallest unit of publishable material). Anyway, if you've read this far, I applaud you. I kind of went off on tangents. But my entire point is that the high school system around here is tailored to a certain group, and everyone outside of this "box" are essentially outcast. @Thane Plambeck AP exams aren't very good, I don't think. They make teachers teach "to the test". So in the end, things aren't taught, they're drilled. Voluntary CLEP exams are a much better idea I think (which can be taken almost any time), and don't have this whole air around it like AP exams do.