steven0461 comments on Mechanics without wrenches - Less Wrong
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Have there been studies of how worker productivity is distributed? We should at least be able to get economy-wide income data, which gives us nearly the same info if we assume peoples' pay tracks the value they add.
In math, it does feel as though one's mathematical power is something like exponential in the amount of time one has spent on solid math study, at least until one hits the frontiers of one's subfield. Algebra I takes a year to learn, but a few years later, the content of algebra I seems similar to the content of a section (not even a chapter) within a semester's course. I suspect similar increases in my ability to learn other competencies as I "learn how to learn" those fields, but it is harder to quantify. Do any of you programmers, or others, care to estimate how your productivity has changed with focused efforts to learn, or to learn how to learn?
Math feeds on itself. It takes a simple concept, and then examines it from a dozen points of view, seeing more and more structure in the idealized problems, both on formal and intuitive levels. As a result, having learned some math, you can more easily anticipate new structure in the new math, and in other problems, as on the intuitive level, it generalizes very well because of the simplicity of constructions that get studied.
Educators must know a great deal about the effect of mathematical sophistication. It's interesting how undergrad textbooks, even on the subjects I know nothing about, seem boring and longwinded, and it's often more instructive to just find a tutorial paper, tapping greater depth through keywords and references.