You might have a quite extensive collection of mathematical texts on one of you shelves, but have you completely read through any of them? While still in lockdown and with the spring semester over, I’ve had a lot of time to read through my shelf. When reading textbooks I primarily use two strategies, and I’d like to share these in the following.
When it comes to studying I spend a lot of time optimizing my strategies beforehand. I don’t want to spend hours reading a textbook if I’m not going to get much out of it, and I used to think that this meant reading the textbook line by line to maximize my understanding. It took me a long time to figure out that this isn’t always the best idea especially with subjects I’m not familiar with. The two strategies I use are deep reading and immersion reading, and they are both beneficial in different ways.
Deep reading is an improved version of the line by line reading I used to do (mentioned above). It involves reading the text in passages in the following manner:
- Read line by line.
- When you come to a line, statement, phrase, etc. that isn’t clear to you, treat it as an exercise, solve it, and make a note to remember how you solved it (usually recorded in the text). Then move to the next line.
- Once you’ve done the above two steps for the entire passage, reread the passage again without stopping.
The idea here is that you’re trying to understand all the details in a passage and then reread the passage for a wholistic understanding. In this respect, deep reading is like reading with exercises along the way. It is not uncommon for authors to reference exercises in text and I will usually take it upon myself to solve these exercises when they are needed. The method of reading is very slow (at the very least you’re reading everything twice), and I reserve it for when I am going to be parsing through a text over an extended period of time. Most commonly, this is for textbooks used in courses. For some fairly extensive (incomplete) examples on how I take notes for deep reading, see the following:
- Notes for number theory in function fields – Rosen
- Notes for algebraic number theory – Neukirch
- Notes for Differential Forms in Algebraic Topology – Bott & Tu
Immersion reading is vastly different from deep reading, but a strategy I implore much more often and is still very effective. It is in the same spirit of learning a language via immersive study. With this method you’re only trying to take in the general idea of the theory and intuition behind proofs rather than details. Essentially, all you do is read the textbooks like you would a novel, but stop for just a minute at things you don’t immediately see how to understand and then move on. Don’t pause for more than a minute and don’t write anything down (marking is ok) or else you’ll be deep reading. It might feel like you’re not understanding anything at times, but don’t let this discourage you! I’ve found that as long as you’re understanding around 10% of the material you’re benefiting. The idea here is that you’re gaining familiarity with the material just by letting it steep in your head.
I tend to immersion read a textbook when I have a week or two where my workload is lighter and I always pick a text that I know fairly well (i.e., I’ve taken a course or two in). I also often immersion read texts again after a span of time where I have significantly improved my mathematical understanding. This might be overkill in some respect (but I quite like reading textbooks in my free time). Usually I’ve learned a few things that makes the immersion reading easier and I end up forming a better general picture of the theory than I had before.
My Mathematical Library
To conclude, it would only be fitting to include a photo of my mathematical library (currently moved to a table instead of a shelf):