You sometimes run across this expression "knowing experimentally" in Quaker literature, and I latch onto it, thinking in part of Guinea Pig B, Bucky Fuller's sometime vision of himself.
I'm Guinea Pig K then; we each get "a self" to test the world with (a philosophical notion). Maybe it's a "virtual self" in some way, or, in the Buddhist sense, it's co-defined by the world. When our sense of the world changes, so too does our sense of self, and vice versa. Wittgenstein's Tractatus is a lot about this.
What puzzles me is when educators object to our planning around FOSS-based and/or "object oriented mathematics" with complaints that it hasn't been tested sufficiently.
There's a chicken-egg problem here, in that you don't get to skip the experimental part. Strong schools encourage trying new approaches, reward initiative, as only by testing new ideas in the field will we gain in experience.
In any subculture that innovates, you need the equivalent of "test pilots", people who'll "boldy go" (as they say on Star Trek). "Guinea pigs R us" in other words.
I've put a fair amount of stress on my Saturday Academy pilots, which you'll find written up in scattered posts within these blogs. Here were my tests, my results. Others could copy. I shared lots of lore, not just technical stuff. I made my source code freely available.
And so, that's a wrap in a lot of ways, one guy doing what one guy might do.
In terms of a circus, setting up a tag team of "gnu math teachers": it's a fun idea and would likely help galvanize our economy to new heights. The goal is to boost enrollment in STEM subjects: science, technology, engineering and math.
This isn't at the expense of other subjects so much as a matter of more successfully converging them. Philosophy used to have some responsibility for achieving a good mix.
Sometimes I think the Quaker schools might be up for more experiments of this nature, starting with leadership in some of the flagship colleges. For all I know, that's what a goodly number are trying already.
In the meantime, the existing math track has a lot of problems i.e. we know experimentally that it's broken. But sometimes staring at something broken is more fascinating than coming up with new solutions.
Fighting fatalism is the name of the game.