Math academia is weird to me. In physics, researchers insist that there are applications to even esoteric problems, often to a fault. Mathematicians,on the other hand, often get offended at the idea that math needs to be created to solve other problems, that rather, math is done because it is elegant and beautiful, and if it happens to be useful to someone outside math that's just an unintended side effect.
The author says doesn't understand his own thesis, five years later. It's probably reasonable to suggest that most other PhD mathematicians don't understand his thesis, either. Probably the only people who do understand it would be his advisers or people in that research group trying to build on his work. Maybe, ten years from now, no one will be able to read his thesis at all, and it will be forgotten.In math, this isn't seen as such a tragedy because of the differences in motivations like I mentioned above. I believe this happens in other sciences, too, but people in those fields don't like to talk about that. The idea that the average PhD dissertation in physics or biology in incomprehensible to its author five years later would make people question why anyone bothers supporting them, because people fund physics and biology expecting to get something out of them.
The net product of all the resources put into this PhD will be a guy who claims he is better at problem solving and maybe a marginally successful online math tutoring website. It may be that having more people who are good at the abstract concept of problem solving is a good thing for society, but I'm skeptical that this will be easy to show, even though I'm not opposed to the idea. Problem solving is a useful skill that some people develop from their education. But the problem is that it isn't easy to define or measure what we mean by "problem solving." As a result, it's hard to implement an educational policy that increases "problem solving"and it's basically impossible to evaluate whether or not the policy worked.
Curriculum standards are based on notions of what students should know (with some movement towards performing discrete acts of reasoning, itself very limited). Assessment is dominated by an obsession with short-term knowledge gains.
This sort of claim is made by people who live in the sheltered part of academia, who don't understand many of the motivations behind assessments,and who don't understand just how bad the state of education is. It's very tempting to look at the ideal of the system we have now, and try to examine how we can optimize that. This optimization is aimed at making the experience better for the best students at the best schools, so that they can find education more fulfilling. However, those sorts of improvements are superficial when compared to the real problems of education. What material students are learning isn't the problem, the problem is that students aren't even learning the basic knowledge that we're teaching them! It isn't primarily a problem of educational philosophy, it's a problem of educational application.
If we look at the best schools,students do learn basic knowledge and do well on standardized tests. I went to a fantastic public school, and we all thought that the standardized tests were stupid easy. Even our "dumb" kids, who didn't get good grades, still managed to score fairly highly on the tests, because the tests were about three grade levels below even our slowest students. These assessments have incredibly low standards: they don't expect you to be able to do any more than basic algebra (solve for x level) by the time you graduate high school. The reason for having assessments with these low standards is solely to determine the schools where something is going wrong. Since so many schools do fail these assessments, something is going very, very wrong indeed.
The detail left out of this article is how much more difficult it is to teach problem solving than basic knowledge. We've got basic knowledge pretty well figured out in the best schools, so we know that it can work in a proper environment with motivated, wealthy students and skilled teachers. If some of these schools tried to teach problem solving or philosophy or critical thinking or whatever buzzword, it probably wouldn't go terribly and at the very least it likely wouldn't get in the way of their basic knowledge. But can you imagine trying to implement these sorts of policies in a troubled school, where students can't even be taught algebra properly? It would cause utter confusion, with neither students or teachers being able to comprehend what was going on or why it was being inflicted upon them.
Teaching problem solving is hard, and developing it demands the best teachers and the most motivated students. Without that, any attempt will result in complete failure. You can't half-teach problem solving skills like you can half teach algebra or american history, you either teach them, or you recite a bunch of gibberish at students that they then recite back, and no one learns anything. It isn't a simple fix, and it isn't the most pressing problem in education when we have schools where students have 12 years of math and still can't solve for x.