> Normally going from no calculus to the multivariable chain rule as applied to differentials or as a best linear approximation takes at least three semesters in university.
Multivariable chain rule is much simpler than the 18 months learning curve would imply. It boils down to figuring out what a function is, what a derivative is, what the chain rule is and generalizing to multiple dimensions. We could probably teach it to a sufficiently logically apt high-schooler within a week, provided we could find a sufficiently motivating use-case.
The lack of compelling use-cases being the other major obstacle in learning math. Why bother rote memorizing tens of concepts and hundreds of factoids, when a lot of math texts pride themselves of building the perfect theory in abstract, decoupled from the original motivations.
I personally don't find a lot of motivation to learn calculus because engineers in the 19th century needed to figure out how to make better steam engines or cannons or because physicists wanted to understand electromagnetism. My own motivation was: okay, neat, derivatives. This says a lot about a function! What happens now if we try to do this with more variables? Oh, wow, look, the chain rule gets all weird now and grows additions it didn't have before!
For other people, I guess you need to find a different motivation. Maybe neural networks will do it for some. I must admit that I picked up a neural networks text in 1995 because I wanted to build robots, didn't understand a word of it, and ten years later I got a degree in mathematics having long ago forgotten about the neural networks book which I only recently picked up again. But regardless, ever since I was a little kid, mathematics is just something that naturally attracted me.
We should not minimise the intrinsic interest of the subject itself either. There is an artistic side to mathematics, where we do it because it's beautiful for its own sake. Not all mathematics needs a purely practical reason to justify its study.
> Multivariable chain rule is much simpler than the 18 months learning curve would imply. .... We could probably teach it to a sufficiently logically apt high-schooler within a week, provided we could find a sufficiently motivating use-case.
Supposing this were the case, given that there's no shortage of homeschooling and "alternative" high schools out there, can you find a case where this has actually happened.
Multivariable chain rule is much simpler than the 18 months learning curve would imply. It boils down to figuring out what a function is, what a derivative is, what the chain rule is and generalizing to multiple dimensions. We could probably teach it to a sufficiently logically apt high-schooler within a week, provided we could find a sufficiently motivating use-case.
The lack of compelling use-cases being the other major obstacle in learning math. Why bother rote memorizing tens of concepts and hundreds of factoids, when a lot of math texts pride themselves of building the perfect theory in abstract, decoupled from the original motivations.