https://en.wikipedia.org/wiki/Higher-order_thinking
I would be quite happy to see these kinds of concepts rolled into one.
Looking back, most times I heard "higher order thinking" at school, I feel like they meant "thinking at a higher level of abstraction" (as I would describe it now).
You could make an argument that the kind of "nth derivative thinking" identified here is similar though.
For m variables, there are m first orders, m^2 second orders, etc. and most proofs about, eg convexity, amount to a proof over that shape.
The reason that n-th order reasoning about an m-dimensional problem is difficult is that it requires reasoning using m^n objects.
If you figure we can remember about ten things intuitively, that’s just enough for three dimensions and two derivatives — enough to do motion analysis.
But it’s leagues away from being able to do the second order of a ten-dimensional problem intuitively.