calculus is required to understand classical mechanics, but intuitions about classical mechanics can inform and accelerate how you learn calculus. I actually tutored calc 1 a little bit in college and most of the time the people I was helping could do the calculations asked of them fine, but felt lost and confused because they didn't know what the output actually meant. everyone learns linear algebra before abstract algebra, but even a little background in the latter creates many opportunities to think "ohhh, this is just like X!" that make the former easier to pick up--and when you revist the latter, you'll probably understand it better too because of the connections you made
is it better to learn python or java to build a cursory understanding of programming, and then c and x86 to see what "really" happens "under the hood?" or the other way around, starting from base operations and then layering abstractions on top? I don't think one is strictly better than the other. when I first learned sql, joins were intuitively obvious to me but bewildered the person I was learning with, despite the fact that he'd been in IT for years and I barely knew how to program, because I happened to know set theory. I wonder what it would be like to do the traditional algorithms and data structures stuff before ever learning to program. it might make picking up a lot of details easier! I wonder why we don't teach children logic gates and binary arithmetic before they learn decimal. it's actually simpler!
in general I don't think there's a good reason why you must teach raw technique before practical application, or a concrete special case before a broader abstraction. even an impefect understanding of something "higher" can give you hooks to attach information about something "lower." familiarity and comfort and an understanding of interrelationships between knowledge is so much more valuable than perfectly executing each step on a given track
when I want to get acquainted with a new field, I usually start by reading current research, barely understand any of it, and then work backwards through the author's previous papers and the papers he cites habitually. my cursory grasp on the latest version makes it easier to understand earlier sketchier versions, and at the same time the development history of the older shines light on the newer. sure you can always "start with the greeks" as it were and work your way up, but I don't think this is objectively better than going the opposite direction
really I think of knowledge as more a highly interconnected graph than a series of linear tracks. as long as you can anchor something somewhere it's valuable to pick it up. it can reinforce other things near it and you can strengthen your understanding of it over time. and getting into the practice of slotting arbitrary things together like this is good training for drawing novel inferences between seemingly disparate topics, which I think is the most valuable form of insight there is
While I completely agree with your general comments, I still want to emphasize how hard this ideal is to implement in practice.
Yes, every subject is linked to every other subject. If I'm tutoring introductory physics, I will use examples and analogies from math, computer science, engineering, biology, or more advanced physics, depending on the background and taste of the student, and it works fantastically. But if I'm lecturing to a crowd, this is impossible, because the students will differ. If I draw a link, some people will think it's enlightening, some will think it's totally irrelevant, some will think it's boring, some will think it's so obvious it's not worth saying, and most will just get more confused.
The same thing goes for the top-down "working backwards from cool results" approach; it's supposed to bottom out in something you know, but whenever you teach multiple people at once, everybody knows different things. The bottom-up linear approach is useful because it gives you a guarantee that you can draw a link. If I'm teaching quantum mechanics I expect to be able to lean on intuition from classical mechanics and linear algebra. If I didn't know the students had that, I would draw a lot fewer links, not more.
Similarly, "if people learned X in school, then Y would be easier to understand later" is true for almost any values of X and Y, because of the interconnectedness of knowledge. But if you ask any math teacher, they'll tell you the school curriculum is already bursting at the seams. You can't just add logic and set theory to existing school math without taking something out. In the 70s we tried taking out ordinary arithmetic to make room for that. It was called New Math, and everybody hated it.
is it better to learn python or java to build a cursory understanding of programming, and then c and x86 to see what "really" happens "under the hood?" or the other way around, starting from base operations and then layering abstractions on top? I don't think one is strictly better than the other. when I first learned sql, joins were intuitively obvious to me but bewildered the person I was learning with, despite the fact that he'd been in IT for years and I barely knew how to program, because I happened to know set theory. I wonder what it would be like to do the traditional algorithms and data structures stuff before ever learning to program. it might make picking up a lot of details easier! I wonder why we don't teach children logic gates and binary arithmetic before they learn decimal. it's actually simpler!
in general I don't think there's a good reason why you must teach raw technique before practical application, or a concrete special case before a broader abstraction. even an impefect understanding of something "higher" can give you hooks to attach information about something "lower." familiarity and comfort and an understanding of interrelationships between knowledge is so much more valuable than perfectly executing each step on a given track
when I want to get acquainted with a new field, I usually start by reading current research, barely understand any of it, and then work backwards through the author's previous papers and the papers he cites habitually. my cursory grasp on the latest version makes it easier to understand earlier sketchier versions, and at the same time the development history of the older shines light on the newer. sure you can always "start with the greeks" as it were and work your way up, but I don't think this is objectively better than going the opposite direction
really I think of knowledge as more a highly interconnected graph than a series of linear tracks. as long as you can anchor something somewhere it's valuable to pick it up. it can reinforce other things near it and you can strengthen your understanding of it over time. and getting into the practice of slotting arbitrary things together like this is good training for drawing novel inferences between seemingly disparate topics, which I think is the most valuable form of insight there is