I suspect you and GP are talking about slightly different things. GP is probably more opposed to artificial compartmentalizing of things. As an example:

> You do need to understand calculus before classical mechanics

Yes, but how much calculus? Do you need all of Calc I, II and III before even attempting classical mechanics? And should calculus even be treated independently of classical mechanics?

There are various traditions when it comes to teaching these subjects, and the tradition in the US involves keeping a strict distinction between these things, in addition to a "theory first" approach. Other people have studied things in a different manner. Some of my physics professors from the UK had studied most of the math they knew only as needed when they would get to relevant topics in physics - including differential equations, all of analysis (complex or real), some of Calc III, etc.

Even amongst mathematicians, it was common in parts of Eastern Europe to focus on a problem, and learn whatever theory is needed to solve that problem. They didn't learn theory and apply to problems - they took a problem, and learned whatever theory is needed to solve them. I recall picking up a Kolmogorov textbook on analysis and being surprised by seeing this approach, along with the informality with which everything is discussed.

And just a minor quibble:

> classical mechanics before quantum mechanics,

You don't really need to know much except the basics. I think the classical mechanics we covered in our typical "Engineering Physics" courses was sufficient to dive into proper quantum mechanics. It's nice to have been exposed to Hamiltonians in classical physics prior to taking QM, but really not needed. There's a reason neither schools I attended made the classical mechanics courses as prereqs to QM. In fact, I would argue we should split things up a bit: Have a course to teach the very basics of energy, momentum, etc. Then make it a prerequisite to both classical mechanics and quantum mechanics.

>And should calculus even be treated independently of classical mechanics?

Thank you. The value of calculus didn't really click for me until the next year in physics, at which point I wished I had paid more attention.

It's really unfortunate that you first have to drill high-speed high-precision symbol manipulation, devoid of any meaning, for so long before getting any indication of why you'd want to be able to do it.

I like Lockhart's analogy: mastering musical notation, transposition, etc. before being allowed to play your first note.

https://www.maa.org/external_archive/devlin/LockhartsLament....

Some people like to work their way in reverse. I'd often pick a really complicated subject I'm after like "stellar fusion" and then work my way downwards and learn whatever I need to learn in order to understand it. If I had to start from differential mathematics, without knowing why I need it, I'd probably give up.

This is the big problem I had with engineering in school, so much was based on faith that you needed it. And just now as I'm writing this, I realized that's completely counter to my personality.

side note: I remember Einstein needed to learn particular math skills to help prove his ideas and sought out to learn it.

Are there any notable teachers who take or have taken this approach?

I can't think of any from my education, everything was bottom up building. But when I self-teach (as it naturally emerges from pursuing projects related to work or hobbies) it's often top down digging.

It didn't work for the GP. That makes it poisonous and evil.

I've seen this sentiment way too much on HN. X didn't work for me, therefore X is a scam, its perpetrators are evil sociopaths, and if it worked for you you're a cog in the machine, man.

I'm a believer in individual learning styles, and hard as it is for me to understand personally, I do recognize that lots of people may learn best by lecture and drill. I certainly don't think bad of them for it. but many more people are not suited for this, and forcing them through the education system as it exists does them great harm, and I empathize strongly with them

Lecture and drill has been out of vogue for decades. The hot thing is active learning and high-impact practices. And frankly it has the opposite problem of dragging the class down to the slowest common denominator, wasting the opportunity to teach more to the high-achievers.

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.