How to efficiently read mathematically and theoretically dense books in STEM fields?

Question

When it comes to reading, there are literally thousands of methods from Speed Reading to SQ3R to Sequential(Word by Word till the end). My question is regarding reading mathematically/theoretically dense books as a graduate student. My question is primarily targeted to Science, Technology, Engineering, and Math (STEM) fields.

I have read the other question on SE and this is designed to act as a question on similar lines but for books (> 400 pages) typically found on Reading Lists for Quals. I believe how one reads a book differs significantly from how one reads a paper. (This could be a question as well but IMO, the length, intention and structure are sufficient to cause the difference)

In order to make this an objective question rather than a vague and open ended one, I wish to concentrate on the following:

1. Should a book be read from start to finish word by word or through iterations (Skim, Analyse, Summarize)?

2. If I am interested in a particular chapter with a lot of dependencies, is it in my interest to read everything till that chapter or read that chapter > google unknown terms > read chapter again and loop?

3. If one gets stuck for over a certain threshold at something is it wise to continue assuming it as true or to persevere till the end and figure it out. This is true for research papers, is it true for books?

4. How much time per (mathematically dense) page is ideal? This will vary a lot with field but not so much with person as it would with fiction (IMO).

Answer 1

I have never (as far as I can remember) read a technical book cover to cover from start to end. Most of the time, I have a problem in mind that I want to solve, and I'm looking for tools to solve it; more often than not, if I'm trying to learn out of a book, I'm actually reading three or four books at once. I dive into the middle of a book that seems most relevant; if I don't understand something, I'll back up, and if I don't understand that, I'll back up again, and if I get really stuck I'll put the first book down and pick up a more elementary book, and so on until I'm on firm ground again. ("Getting really stuck" only happens after relying to work through/reconstruct details on my own, in addition to trying to understand them from the book's presentation. I have taken months to read through one page, always feeling just close enough to understanding that I never felt "really stuck".) Whenever possible I pop back up the reading stack with my target problem in mind, skipping entire chapters if they don't seem relevant (but backtracking if I discover later than I'm wrong), working forward again until I either find the tool that I'm looking for, conclude that I've been on a wild goose chase, or give up on the book.

Yes, I miss a lot this way. Yes, I get a lot of weird ideas that I later have to kill off. But I just don't have the patience to read large volumes of technical material that doesn't seem at least remotely relevant to some problem at hand, and prioritizing often leads me fairly quickly to tools that work.

Your mileage may vary. Caveat lector.

Answer 2

The types of books you refer to are unlike publications, in that they typically detail the results of research that occurred 5+ years ago. As such, if you're reading such a book, it's typically for one of two reasons:

1. To be brought up to speed on a topic with which you're unfamiliar
2. To learn a specific technique which is discussed in the book

In either case, you'll want to typically be interested in only a subset of the book. I would recommend reading through that section slowly, section by section, as you would a paper. It will probably take a while. I have never read an academic book cover-to-cover. Do note that your speed-reading skills will likely be less useful here, as STEM literature typically does not lend itself to speed reading. There is no "ideal time" that can be stated. When you begin, a single page may take you hours. There was one paper I read where a single equation took my lab mate four months to work through. As you get more experience, you'll speed up. Time is a commodity, but information is a more precious one. Spend the time necessary to learn the topic, and especially at the beginning, measure your progress over days, not hours.

If I am interested in a particular chapter with a lot of dependencies, is it in my interest to read everything till that chapter or read that chapter > google unknown terms > read chapter again and loop?

There are two approaches here. The one you mention - google unfamiliar concepts - can work well. However, oftentimes the author of the book will mention exactly where the dependency is (e.g., "As we discussed in the previous chapter, ..."), so you can identify which parts of the book you need to read through. That may be more useful, as any terminology will be consistent within the same book. For some topics (math & engineering in particular), other papers will often use different terminology, needlessly complicating the learning process.

If one gets stuck for over a certain threshold at something is it wise to continue assuming it as true or to persevere till the end and figure it out. This is true for research papers, is it true for books?

This is simply a function of how necessary the topic is. When you get stuck, figure out what concept is confusing you, read up on that topic, and then continue. This process can take weeks, or even months. If you really want to learn the topic, persevere. If you can continue without that bit of clarity, move on to something more productive.

Answer 3

How you read an academic book depends heavily on what you want from it, but one thing that I think is almost universally true is that you generally don't want to bog down on one page/argument/theorem/definition/whatever on a first read. Later content often adds context and motivation to something that may seem mysterious, so if you get stuck for a while on something, it's a good idea to move on and then come back to it later.

My usual experience reading mathematically dense content, which I think is relatively typical, is that as I go along, the level of my understanding steadily decreases, and eventually it gets to the point that I'm learning very little, so I go back to the parts I understood well and start reading again from there, this time understanding a bit more, and I iterate this until I've learned what I set out to learn.