If I had to guess why "math reform" is misinterpreted as "math education reform", I would speculate that school is the only contact that most people have had with math. Like school-physics or school-chemistry, math is seen as a subject that is taught, not a tool that is used. People don't actually use math-beyond-arithmetic in their lives, just like they don't use the inverse-square law or the periodic table.

Teach the current mathematical notation and methods any way you want -- they will still be unusable. They are unusable in the same way that any bad user interface is unusable -- they don't show users what they need to see, they don't match how users want to think, they don't show users what actions they can take.

They are unusable in the same way that the UNIX command line is unusable for the vast majority of people. There have been many proposals for how the general public can make more powerful use of computers, but nobody is suggesting we should teach everyone to use the command line. The good proposals are the opposite of that -- design better interfaces, more accessible applications, higher-level abstractions. Represent things visually and tangibly. And so it should be with math. Mathematics, as currently practiced, is a command line. We need a better interface.

Anything that remains abstract (in the sense of not concrete) is hard to think about... I think that mathematicians are those who succeed in figuring out how to think concretely about things that are abstract, so that they aren't abstract anymore. And I believe that mathematical thinking encompasses the skill of learning to think of an abstract thing concretely, often using multiple representations – this is part of how to think about more things as "things". So rather than avoiding abstraction, I think it's important to absorb it, and concretize the abstract... One way to concretize something abstract might be to show an instance of it alongside something that is already concrete.

The mathematical modeling tools we employ at once extend and limit our ability to conceive the world. Limitations of mathematics are evident in the fact that the analytic geometry that provides the foundation for classical mechanics is insufficient for General Relativity. This should alert one to the possibility of other conceptual limits in the mathematics used by physicists.