> what's wrong with presenting everything in the correct abstraction to start with?
Nobody can retain the material that way. Each thing you learn has to be attached to other concepts you already have, and your skills build on one another.
Nobody teaches Peano's axioms of arithmetic to kindergarteners; they're still learning how to identify shapes, to compare quantities. When they eventually learn even the simplest proofs, they'll build on the basis of careful attention to detail that they learned while mastering basic arithmetic.
Nobody teaches general relativity on the first day of college-level physics class. Students are still learning calculus at that point; even if they already had a calculus course, this is their first opportunity to apply it.
And nobody teaches the fundamental proofs of calculus on the first day of calculus class; you can use sloppy language like "infinitesimals" to establish a good intuition for how to use derivatives and integrals.
If you tried introducing this material from the bottom up, it wouldn't take. But if I'm wrong, sure, go try it and see how it works.
You're right, I shouldn't have said the correct abstraction, but you should at least provide a complete one. You don't have to build from the most fundamental possible assumptions to build from a set of tools that is provably complete for some domain of problems.
Doing otherwise is like giving someone a screwdriver that only turns clockwise.
Also, I challenge the notion that "no one can retain material that way". Have you ever met anyone who has retained the material any other way who didn't do it despite the system?
Essentially, this is impossible for anything that’s not easily explainable all at once- so it’s useless for any advanced domain.
You can’t just give people a firehose of information. When you learn acoustic engineering, you have to first learn the prerequisite math needed to understand later concepts such as room architecture. It simply does not make sense to jam pack it all in at once, because 1) it won’t make sense and 2) nobody has that kind of memory.
Next, if you want to cover all possible use cases, that could take forever in certain domains.
Last, some of this stuff is objective as to what’s necessary, or what cases need to be covered.
No, what’s better is to give general knowledge as needed, and the student can seek out knowledge for various corner cases. It would be ridiculous otherwise.
Nobody can retain the material that way. Each thing you learn has to be attached to other concepts you already have, and your skills build on one another.
Nobody teaches Peano's axioms of arithmetic to kindergarteners; they're still learning how to identify shapes, to compare quantities. When they eventually learn even the simplest proofs, they'll build on the basis of careful attention to detail that they learned while mastering basic arithmetic.
Nobody teaches general relativity on the first day of college-level physics class. Students are still learning calculus at that point; even if they already had a calculus course, this is their first opportunity to apply it.
And nobody teaches the fundamental proofs of calculus on the first day of calculus class; you can use sloppy language like "infinitesimals" to establish a good intuition for how to use derivatives and integrals.
If you tried introducing this material from the bottom up, it wouldn't take. But if I'm wrong, sure, go try it and see how it works.