When to skip sections of a book when self-studying?
When you're taking a mathematics class, you usually know exactly what sections of a book you need to know, and you can focus your time on these important sections.
However, when studying by myself, even when I'm trying to study the book as thoroughly as possible, I often feel tempted to skip sections of material (maybe a subsection of a chapter, a proof, or an exercise set). Yet at the same time, I don't want to skip it, fearing that what I want to skip might be something really important. Some reasons you might want to skip are:
- You might feel that you already know it well enough
- The proof or the exercises might be too difficult or boring
- The section might seem not very important
For instance, when self-studying from Apostol's Calculus, I felt the need to skip the section on calculating errors of the taylor series for the log function because it seemed unimportant, and the 'rigorous' proof of the FTC which seemed completely unmotivated.
What are your strategies for studying material in a thorough, complete way? When is it best to skip -- and how should you determine if the material you're skipping is important or not?
I would recommend the following strategies:
(1) If you are having difficulty knowing which parts of a mathematics textbook to skip and which to read, then try to mostly read "user-friendly" mathematics textbooks for the time being. Usually, there are many mathematics textbooks on a given subject which give a roadmap at the beginning of the book detailing which sections/chapters of the book can be skipped and which are important/interesting. I should remark, however, that sometimes these roadmaps are not entirely accurate and alleged optional sections are actually mandatory to understanding other parts of the textbook. Therefore, it is important to exercise caution, but be assured nonetheless that results used in other parts of the textbook will be quoted when applied and thus you can always return to the optional sections when necessary.
(2) If you are thinking about skipping certain theorems and proofs rather than entire sections or chapters, then I would recommend you to at least read the statements of the theorems which you skip (if you do not read the proof). The reason is that, if the theorem happens to be applied later in the book, then you at least know the statement and can accept it on faith for the purposes of the application. If the theorem is sufficiently important that it is applied several times throughout the textbook, then you might feel guilty and go back and read the proof later. If not, then it was probably a good choice (in terms of time) that you did not read the proof. (I hasten to add, however, that some theorems may not have any applications in the textbook despite being important and interesting results.)
(3) Of course, you can also ask on this website whether or not a particular proof is worth reading or whether or not you can skip certain material! I believe that the totality of all mathematics textbooks read by users on this website is quite a broad and deep collection.
(4) Also, at the basic level, I think it is generally not a good idea to skip too much material. The reason is that experienced authors usually choose the material to include in their textbooks very carefully. For example, Walter Rudin is an author that comes to mind who very rarely includes a result in his textbooks that is not to be used elsewhere in the textbook (unless the result is important and interesting enough to be stated despite not having applications elsewhere in the textbook). In other words, most authors do not include arbitrary material in their textbooks and you should have faith and accept that whatever they do include merits inclusion. (In particular, at least for the time being, I would recommend you to read well-known books or books of well-known (and good!) authors.)
(5) Ultimately, mathematics is an extremely broad and deep subject. You will eventually need to become accustomed to accepting results on faith as there are simply too many results worth knowing and too little time. Therefore, you should get into the habit of applying results whose proof you do not know. I mentioned earlier that if you apply the result sufficiently many times, then the chances are that it is important and it is worth knowing the proof of the result. Of course, you should read a healthy number of proofs as well.
I hope this helps!
First peruse the preface to see if the author has explicitly discussed which sections are optional, or the chapter/section interdependencies. If not, use Google or Amazon books to search the book to see if the omission candidate is invoked elsewhere in the textbook. If not, then its a safe bet that it is probably tangential to the main course. But that's no guarantee that it won't prove essential in later courses. One really needs some expertise to make such judgments, so seek advice from others if at all possible.
As a student I found it useful to draw big graphs of the logical dependencies of theorems. Nowadays there is good graph layout software to aid in such endeavors. Doing so will help you to obtain a better global feel of the logical structure of the theory, to help ensure that you do not miss the forest for the trees.