Well, there are structural problems and personnel problems. Most big, intractable problems come down to a combination of these two -- and the personnel problems

Well, there are structural problems and personnel problems. Most big, intractable problems come down to a combination of these two -- and the personnel problems are from structural causes. The biggest issue is that we have no systematic tracking in the U.S. That means students are not given the opportunity to go to a vocational school, everyone is routed to a "college prep" high school. That causes people who would rather work on more practical things to listen to more abstract things, but watered down to the point of being uninteresting. Math is too boring for the smart kids, and irrelevant for everyone else. The second biggest issue is personnel. Our math teachers aren't very good at math, and tend to be at the bottom of the barrel in terms of SAT scores. In other countries, the brightest college graduates go into teaching, or at least those in the upper quartile. In the U.S., it's those in the bottom quartile. I used to tutor Math education majors in Arizona - of all the majors I tutored, the math education majors were the worst, in terms of their knowledge and enthusiasm for the subject. At that time they needed only a single class beyond the standard freshman math curriculum. It was called "Advanced Calculus", but was basically just an introduction to real analysis. Basically you just go back over single variable calculus and are told the completeness axiom and Cauchy limits, and then you prove the major theorems of single variable calculus. That's it. The Math majors generally took this class in their freshman year (since they would AP test out of calculus, which was the only pre-requisite). The Math Ed majors would typically take it in in their Junior year, because almost none of them took AP calculus and many had to take remedial math classes. And it was a massive struggle for them. I would hear so many math ed majors complain that they hated math that one time I asked one "Why don't you try another major, if you hate math?" and the response was "Oh, I like teaching, but I hate math." That was really a moment of epiphany for me since I looked back on all the terrible math teachers I had, and saw the other side of the coin -- they hated the subject. They were having as much of a bad time as I was. That comes through, and influences the pedagogy. In terms of books, I recommend anything by George F. Simmons, who adds historical notes and examples to his books. For example, "Differential Equations with Applications and Historical Notes".