[mpoteat]

The I in PID is referring to a pretty simple arithmetic sum inside of a control loop. Around 30 lines of simple C.

The original post is presumably referring to symbolic integral calculus, perhaps in a multivariable context if we're referring to partial DEs.

The level of rigor and complexity here is a couple orders of magnitude difference. I think there's still a case to be made for autodidaction of higher mathematics (it was certainly the path I went down), but I think unschooling is probably not best for most folks.

[ultrarunner]

This nitpick is fair. My kids are pretty young still. 30 lines of C is not simple for them without help. 30 lines of C wasn’t simple for the employees of the schools I went to either, who were baffled by the technology that was shoved at them. My “computer class” teacher lamented that she only got stuck with the class because the rest of the faculty realized she had figured out how to use the mouse before anyone else. Simple or not, I would make a small bet that not a single faculty member in that district to this day knows what a PID controller even is, and a very large bet that no one at the time did. They provide a very standard education.

But yes, you are quite correct that a PID controller is not the only (or even most complete) application of integral calculous. The example was given as a pathway by which an unschooled child could become aware of a topic concurrent to an understanding of its importance. By no means did I intend to imply that the education should stop there. There's also nothing stopping an unschooled child from pursuing higher education or degree. Indeed, already knowing how to learn seems to place them at an advantage in such a case.