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People will complain about that too. The introduction is "redundant" or "trivial". The book is a long, bloated slog. (Since what you're proposing will unavoidab

by knzhou 7y ago

People will complain about that too. The introduction is "redundant" or "trivial". The book is a long, bloated slog. (Since what you're proposing will unavoidably increase the page count.) The author intentionally confuses students by alluding to some "main theorem" but waiting a hundred pages to say what it is, instead of putting it in front where it belongs. For every complaint about a textbook, there is an equal and opposite complaint if it had done the opposite thing. At some point we have to accept that the underlying concepts inherently take time and effort to assimilate, that the easiest style to learn from is different for different people, and that the authors really did think about all these issues when they were writing. They clearly care about exposition, otherwise they would not have written a book in the first place, a thankless and financially useless task.

LolWolf 7y ago
I mean, perhaps; though I've found it does quite the opposite. The idea is proposed front and center, sure (this approach does not preclude that), but far less time is spent explaining the motivation and doing mathematical yoga to clarify steps in a proof than is spent providing clear examples and sensible constructions.[1] I am not, of course, saying that all textbooks (or papers) will be better served by this structure, but I think a surprisingly large amount are. This may also be a consequence of the field I'm in, as well, where most statements I've made have several natural generalizations which are all relatively straightforward to prove, but presenting them all under one large umbrella with a very general construction makes the end result much harder to understand. Either way, both ways have their merits, but I've (along with several of my peers and editors) found one much harder to work with and read than the other. Though, I will again mention that this may be due to the specific fields I work in. -- [1] I want to be clear that I'm also not advocating for the "big reveal finish" which many papers (and some terrible textbooks) fail miserably at. Neither of these mediums is a novel. But I am saying that some approach motivating the construction and building the picture in a natural way (instead of a here's the theorem, let's explain why everything is a trivial corollary of it) lends itself to more clear expositions. Perhaps either presentation can be made clear, but I've found it's very, very difficult to make the latter work in a way that readers can easily understand, while the former falls almost naturally from most research notes I've written and keeps people more engaged.
Silhouette 7y ago
People will complain about that too. The introduction is "redundant" or "trivial". The book is a long, bloated slog. Some people will, but they probably aren't the target audience for the book, and most people won't. You don't teach kids who are learning to count about 0 by explaining group theory and identity elements.
- knzhou 7y ago
  I promise you that explaining a technical advanced math subject is less obvious than this, and that even students that come in with the same background will strongly disagree over which books are easiest to read. Maybe it's obvious that sketching only a little formalism is the best. But then some of your statements end up vague, and students stumble on practice problems where they really need to understand the details of the formalism. Maybe it's obvious that stating things as simply as possible is the best, even at the cost of generality. But then your statements won't be powerful enough when people try to go beyond the few simple examples you give, and people will feel betrayed. Maybe it's obvious that a historical approach is the best, for motivation. But historical papers are extremely confusing to read. They're full of redundant checks of things we know are obviously true today, and archaic notation and jargon. People will get annoyed at slogging through ten pages of text to confirm something that is obvious in a modern formalism. Maybe it's obvious that a modern, clean formulation is best. But then the definitions are unmotivated, seem to come out of nowhere, there's not enough connection to the historical development, and your book won't be compatible with older books. Maybe it's obvious that no formalism at all is best; you just show how to solve problems concretely. But this turns what could have been a few simple principles into an enormous bag of ad hoc tricks people need to memorize. Textbooks that do this are derisively referred to as "cookbooks". It is not obvious how to write a book.
  - Silhouette 7y ago
    It is not obvious how to write a book. I agree, and I also agree that readers will differ in which books they find easiest to read. Different people learn in different ways. However, in some respects, it is obvious how not to write a book. Mathematics is not magically exempt from effective technical writing skills, nor all we have discovered about cognitive psychology and how people learn, nor anything else that goes into creating a generally good presentation.

by knzhou 7y ago

LolWolf 7y ago

I mean, perhaps; though I've found it does quite the opposite. The idea is proposed front and center, sure (this approach does not preclude that), but far less time is spent explaining the motivation and doing mathematical yoga to clarify steps in a proof than is spent providing clear examples and sensible constructions.[1] I am not, of course, saying that all textbooks (or papers) will be better served by this structure, but I think a surprisingly large amount are. This may also be a consequence of the field I'm in, as well, where most statements I've made have several natural generalizations which are all relatively straightforward to prove, but presenting them all under one large umbrella with a very general construction makes the end result much harder to understand. Either way, both ways have their merits, but I've (along with several of my peers and editors) found one much harder to work with and read than the other. Though, I will again mention that this may be due to the specific fields I work in. -- [1] I want to be clear that I'm also not advocating for the "big reveal finish" which many papers (and some terrible textbooks) fail miserably at. Neither of these mediums is a novel. But I am saying that some approach motivating the construction and building the picture in a natural way (instead of a here's the theorem, let's explain why everything is a trivial corollary of it) lends itself to more clear expositions. Perhaps either presentation can be made clear, but I've found it's very, very difficult to make the latter work in a way that readers can easily understand, while the former falls almost naturally from most research notes I've written and keeps people more engaged.

Silhouette 7y ago

People will complain about that too. The introduction is "redundant" or "trivial". The book is a long, bloated slog. Some people will, but they probably aren't the target audience for the book, and most people won't. You don't teach kids who are learning to count about 0 by explaining group theory and identity elements.

knzhou 7y ago
I promise you that explaining a technical advanced math subject is less obvious than this, and that even students that come in with the same background will strongly disagree over which books are easiest to read. Maybe it's obvious that sketching only a little formalism is the best. But then some of your statements end up vague, and students stumble on practice problems where they really need to understand the details of the formalism. Maybe it's obvious that stating things as simply as possible is the best, even at the cost of generality. But then your statements won't be powerful enough when people try to go beyond the few simple examples you give, and people will feel betrayed. Maybe it's obvious that a historical approach is the best, for motivation. But historical papers are extremely confusing to read. They're full of redundant checks of things we know are obviously true today, and archaic notation and jargon. People will get annoyed at slogging through ten pages of text to confirm something that is obvious in a modern formalism. Maybe it's obvious that a modern, clean formulation is best. But then the definitions are unmotivated, seem to come out of nowhere, there's not enough connection to the historical development, and your book won't be compatible with older books. Maybe it's obvious that no formalism at all is best; you just show how to solve problems concretely. But this turns what could have been a few simple principles into an enormous bag of ad hoc tricks people need to memorize. Textbooks that do this are derisively referred to as "cookbooks". It is not obvious how to write a book.
- Silhouette 7y ago
  It is not obvious how to write a book. I agree, and I also agree that readers will differ in which books they find easiest to read. Different people learn in different ways. However, in some respects, it is obvious how not to write a book. Mathematics is not magically exempt from effective technical writing skills, nor all we have discovered about cognitive psychology and how people learn, nor anything else that goes into creating a generally good presentation.