> the reason the textbooks are written like that is because that's what was helpful to the authors when they were learning
The author writing a book after 30 years of learning, thinking, talking with other people cannot easily reconstruct what was helpful and what wasn't. Creating 1-dimensional representation of the state of mind (which constitues "understanding") is a virtually impossible task. And here algebraic formalism comes to the rescue. "Definition" - "Theorem" - "Corollary" structure looks like a silver bullet, it fits very well in a linear format of a book. Unfortunately, this format is totally inadequate when it comes to passing knowledge. Very often, you can't understand A before you understand B, and you can't understand B before understanding A - the concepts in math are very often "entangled" (again, I'm talking about understanding, not formal consistency). You need examples, motivations, questions and answers - the whole arsenal of pedagogical tricks.
Some other form of presentation must be found to make it easier to encode the knowledge. Not sure what this form might be. Maybe some annotated book format will do, not sure. It should be a result of a collective effort IMO. Please think about it.
BTW, this is not a criticism of LADR book in particular. The proofs are concise and beautiful. But... the compression is very lossy in terms of representing knowledge.
> "Definition" - "Theorem" - "Corollary" structure looks like a silver bullet, it fits very well in a linear format of a book. Unfortunately, this format is totally inadequate when it comes to passing knowledge.
I really can't emphasize enough that this is exactly how I learn things. I don't claim to be a majority! But saying that no one can learn from that sort of in-order definition-first method is like saying no one can do productive work before 6am. It sucks that morning people control the world, but its hardly a human universal to sleep in.
> Some other form of presentation must be found to make it easier to encode the knowledge. Not sure what this form might be. Maybe some annotated book format will do, not sure. It should be a result of a collective effort IMO.
I 100% agree. Have you seen the napkin project? I don't love the exposition on everything, but it builds up ideas pretty nicely, showing uses and motivation mixed in with the definitions. I've been trying to write some resources of my own intended for interested laymen, so more focus on motivation and examples and less on proofs and such. I like the challenge of trying to cut to the core of why we define things a certain way- though I'm biased towards "because it makes the formal logic nice" as an explanation.
The author writing a book after 30 years of learning, thinking, talking with other people cannot easily reconstruct what was helpful and what wasn't. Creating 1-dimensional representation of the state of mind (which constitues "understanding") is a virtually impossible task. And here algebraic formalism comes to the rescue. "Definition" - "Theorem" - "Corollary" structure looks like a silver bullet, it fits very well in a linear format of a book. Unfortunately, this format is totally inadequate when it comes to passing knowledge. Very often, you can't understand A before you understand B, and you can't understand B before understanding A - the concepts in math are very often "entangled" (again, I'm talking about understanding, not formal consistency). You need examples, motivations, questions and answers - the whole arsenal of pedagogical tricks.
Some other form of presentation must be found to make it easier to encode the knowledge. Not sure what this form might be. Maybe some annotated book format will do, not sure. It should be a result of a collective effort IMO. Please think about it.
BTW, this is not a criticism of LADR book in particular. The proofs are concise and beautiful. But... the compression is very lossy in terms of representing knowledge.