We can waste time in diagnosis when we should be applying a remedy.
We can go astray by defining a project in a wholly negative way, with nothing to offer beyond mere opposition to a perceived adversary.
With the pitfalls above in mind, let me begin by emphasizing the positive content of this website and the two books it advertises, A Brief Quadrivium and Teaching the Quadrivium. I am offering a specific course of study that I consider good and important in itself, without respect to the circumstances in which we happen to live, or the bad habits we must overcome.
Relatively little needs to be said, then, about possible defects of current mathematics pedagogy in the United States. And before any such defects are mentioned, we must acknowledge with gratitude that there are many generous and competent teachers, and many excellent successes already underway in thousands of classrooms.
Still, there are some institutional and cultural features that make it difficult to acquire a proper liberal education in mathematics.
Perhaps the clearest evidence that something is awry is this. Many mathematics departments at colleges and universities in the United States offer a course with a title along the lines of "Introduction to Proofs." These courses are offered to mathematics majors midway through their studies, and are intended to prepare them for upper-level mathematics courses. That is far too late to begin learning proofs. What have the students been doing until then? This very structure reveals that the standard pedagogy of algebra, algebra, calculus, calculus, is giving students habits that must be deliberately displaced before higher studies in mathematics are possible.
Why not start doing real mathematics right away?