A strategy I found works well for deep mathematical understanding and building intuition is cutting off the author, for lack of a better term. For example, when reading a proof, at as many points as you can, stop reading and try to continue onwards from the author’s proof, or try to brainstorm what this new information or finding could be used for. Then read ahead and verify with what was written in the text. If you had an appropriate proof/solution, that’s great. But, if you had a misunderstanding or not sufficient-enough proof, that’s also great. You can now see the fault in your reasoning and build understanding for problems in the future. This is the opposite of passive learning, and it does take longer, but I find it worth it if I am dealing with something I want to know deeply.</p>
I will try having the questions for the specific chapter i’m on side by side with what I’m reading and trying to solve the questions as soon as I feel comfortable solving them. This allows for active reading and prevents me from forgetting topics in a longer chapter. I did this while doing hw, but that was mainly so that I could finish the assignment as soon as possible.
I’ll use both methods while reading Serge Lang’s Grad Algebra and Atiyah & MacDonald’s Intro to Commutative Algebra and provide updates below.