You don’t necessarily have to start from the bottom, though; you can drill down and get to first principles instead of starting without context at low level abs

You don’t necessarily have to start from the bottom, though; you can drill down and get to first principles instead of starting without context at low level abstractions. Both of the angles you two are talking about (providing context and working through the mechanical applications vs explaining fundamental principles that makes things work) seem addressable in the same course if you start with understandable problems and drill down until you get to the principles that allow you to solve those problems. I think it would be quite difficult to do in a reasonable amount of time, but using history as a course outline would be a cool way to explain the motivations/context behind the principles being discussed. That might better illustrate how math research works by explaining the gaps/interesting questions that existed prior to certain ideas and how those ideas evolved and lead to other gaps/questions. The type of math courses I’ve gotten a lot out of already kind of did that, they just usually used more of a condensed, logical narrative than a historically based one.