I’m interested in heuristics for assembling or specific suggestions for a ruthless course aimed for giving a well-rounded math education (that doesn’t trade thorough understanding of a diverse set of tools for better expertise in any particular area).
For example, I find the following techniques useful:
Focus on the simplest texts I don’t yet understand well, not on the hardest texts I can follow
Assemble a list of leading US and UK schools and make sure that I’ve considered topics and textbooks mentioned in their curricula
Given a text, find related texts with Amazon’s “Customers also bought these” lists, in Amazon reviews, with web search
Don’t miss the “gems”, which are often not mentioned in standard courses, but can be found on blogs and forums
Look for lists of recommended books (there are surprisingly few of such lists that are of any value)
What makes specific recommendations valuable for me:
Particularly good texts that may be absent from standard curricula, such as Pierce’s “Introduction to Information Theory”, Courant & Robbins’s “What Is Mathematics?”, Stillwell’s texts, Needham’s “Visual Complex Analysis”, Hilbert’s “Geometry and the imagination”, Arnol’d’s “Mathematical Methods of Classical Mechanics”, etc. (More so for the texts that are not as well-known.)
Texts that are placed in context of other texts that are indicated as being at similar/lower/higher level
Texts with a discussion of prerequisites that names specific other texts and not just topics
Lists of prerequisites that go down a couple of levels without missing lots of intermediate steps, at least within the same topic
And these features make recommendations far less useful:
Assertions of which texts are “better”, where the disapproval turns out to be aimed at books with a different intended audience (what’s “better”, Pinter’s “Book of Abstract Algebra” or Aluffi’s “Algebra”?)
Many alternative suggests, even worse if the “alternatives” are at vastly different levels
Misrepresentation of levels of texts or of order in which the texts should naturally go
Isolated “standard” texts with no context or motivation (a whole list of recommendations can consist of such items)
I’m interested in heuristics for assembling or specific suggestions for a ruthless course aimed for giving a well-rounded math education (that doesn’t trade thorough understanding of a diverse set of tools for better expertise in any particular area).
For example, I find the following techniques useful:
Focus on the simplest texts I don’t yet understand well, not on the hardest texts I can follow
Assemble a list of leading US and UK schools and make sure that I’ve considered topics and textbooks mentioned in their curricula
Given a text, find related texts with Amazon’s “Customers also bought these” lists, in Amazon reviews, with web search
Don’t miss the “gems”, which are often not mentioned in standard courses, but can be found on blogs and forums
Look for lists of recommended books (there are surprisingly few of such lists that are of any value)
What makes specific recommendations valuable for me:
Particularly good texts that may be absent from standard curricula, such as Pierce’s “Introduction to Information Theory”, Courant & Robbins’s “What Is Mathematics?”, Stillwell’s texts, Needham’s “Visual Complex Analysis”, Hilbert’s “Geometry and the imagination”, Arnol’d’s “Mathematical Methods of Classical Mechanics”, etc. (More so for the texts that are not as well-known.)
Texts that are placed in context of other texts that are indicated as being at similar/lower/higher level
Texts with a discussion of prerequisites that names specific other texts and not just topics
Lists of prerequisites that go down a couple of levels without missing lots of intermediate steps, at least within the same topic
And these features make recommendations far less useful:
Assertions of which texts are “better”, where the disapproval turns out to be aimed at books with a different intended audience (what’s “better”, Pinter’s “Book of Abstract Algebra” or Aluffi’s “Algebra”?)
Many alternative suggests, even worse if the “alternatives” are at vastly different levels
Misrepresentation of levels of texts or of order in which the texts should naturally go
Isolated “standard” texts with no context or motivation (a whole list of recommendations can consist of such items)
These are all good ideas and I would be happy to write a post in this direction.