Suggestions for using a new text about a topic on which someone already possess a first course or advanced idea

Question

• The question is seeking the suggestions for using a new text about a topic on which someone already possess a first course or advanced idea. Of course then reading every lines of the text is a shear waste of time.

  However any good text is written from some philosophy and try to show things in a new way. Perhaps that's the reason why even after possessing an advanced concept one can't ever rule out the possibility of learning new things, encountering new problems and realizing a old concept through a new set of lenses from a good text.

  However solving problems, even though the most important part of learning, isn't enough to get the best out of it. In such a situation how should I use a new text? Any suggestion?

---

For suppose I've studied abstract algebra following Gallian. How should I use the Artin text than?

Answer 1

I've done this a couple of times before, and what I usually discover is that I can read the text faster, but there are almost always some points which are new to the text or which I didn't pick up the first time. You very rarely understand material completely the first, second, or third or fourth time. I've learned something new every time I've taught calculus.

But for specific suggestions, I enjoy reading quickly through the new text and doing every exercise that looks hard as well as several problems that look easy, and especially doing every exercise that introduces a new concept. The hard problems are usually ones that get skipped by first courses, and so I find this is the best way to gain knowledge through reading a new book.

If you're not into exercises, I would try using your previous knowledge: every proof you come to, stop and solve yourself without looking. You've seen it before, so you have a good shot, and it will help you know what you already understand and what needs help.

Answer 2

Here is my opinion and experience. I have spent the last year studying algebraic topology. I started with an introductory course at my University that covered the basics (Fundamental Group, Singular Homology/Cohomology etc). Then I did a follow up course on K theory and Characteristic Classes, and now I am learning about generalised cohomology theories and localised K theory.

I have found lately that I can't do a lot of the basic exercises that I once did in the introductory course. So, in an attempt to remedy this, I have been going back and relearning the basics. I find it very satisfying because I already know what the definitions and basic properties are "point toward". That is, I know (at least to some extent) how they will be used in the later theory.

In order to really understand an area of mathematics, I think one should not only remember the definitions, but also understand why those are the required definitions. Then when it comes to Theorems/Propositions, one should try to understand what properties of the objects in study allow one to conclude the Theorem/Proposition.

To this end, when "re-studying" a textbook, I try to think about why things are defined in a certain way, and not in other ways. Also, I try to prove Propositions/Theorem by myself, or at least try to isolate the important properties that constitute a proof. Of course, one can then follow this up by doing exercises, which are intended to do a similar thing; to make the student isolate and understand the important parts of the theory.