what's wrong with presenting everything in the correct abstraction to start with?
Whether you a public school math class, or a git tutorial online, you always get the same thing. A list of procedures for using the tools in the most common cases you are likely to encounter. This seems like a good idea in the move fast and break things ideology, but what do we notice about people's skills in the real world? No one knows how git works, and no one knows any math by the time they leave high school.
Give people a set of tools and prove to them that it solves every problem in some domain. If they can't solve the problems using the most primitive, complete toolset you can give them, then the case where they solve the problem is the edge case.
As an example, often, we will have students write fractions in "lowest terms", then only present students with fractions in which the numerators and denominators only share a few, small common prime factors. But checking prime factors is absolutely the hardest way of solving this problem, and unless you're exhaustive in your search, you can't have any confidence that you actually have solved it.
Students are intrinsically aware that this wold be tedious in general, and it gives them anxiety to know that they could have missed something, or not tried sufficiently large divisors. They have no confidence in their tools, and rightly so. The only reason they can solve these problems at all is because the instructor gives them problems which can be solved this way.
The origin of the question "when will I use this in the real world?" is the instinct that the tool you have been given is only for edge cases, and cannot be relied upon.
I think you are exactly wrong in your assessment. Getting into the weeds is the only way to learn, because almost everything in the world is weeds, some of those weeds just happen to be called crops.
> 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.
Whether you a public school math class, or a git tutorial online, you always get the same thing. A list of procedures for using the tools in the most common cases you are likely to encounter. This seems like a good idea in the move fast and break things ideology, but what do we notice about people's skills in the real world? No one knows how git works, and no one knows any math by the time they leave high school.
Give people a set of tools and prove to them that it solves every problem in some domain. If they can't solve the problems using the most primitive, complete toolset you can give them, then the case where they solve the problem is the edge case.
As an example, often, we will have students write fractions in "lowest terms", then only present students with fractions in which the numerators and denominators only share a few, small common prime factors. But checking prime factors is absolutely the hardest way of solving this problem, and unless you're exhaustive in your search, you can't have any confidence that you actually have solved it.
Students are intrinsically aware that this wold be tedious in general, and it gives them anxiety to know that they could have missed something, or not tried sufficiently large divisors. They have no confidence in their tools, and rightly so. The only reason they can solve these problems at all is because the instructor gives them problems which can be solved this way.
The origin of the question "when will I use this in the real world?" is the instinct that the tool you have been given is only for edge cases, and cannot be relied upon.
I think you are exactly wrong in your assessment. Getting into the weeds is the only way to learn, because almost everything in the world is weeds, some of those weeds just happen to be called crops.