Congratulations on the accomplishments! I did something similar a few years ago, using Spivak’s Calculus rather than Apostol’s (though I did consider using Apostol’s).
I used to think similarly to you, that I needn’t artificially reinforce concepts because those I actually need with be naturally reinforced in the wild, while those I don’t won’t, leaving me with only the skills I actually need.
However, this argument is subtly flawed. It assumes that concepts which are reinforced in the short-medium term are important, and concepts which are important are reinforced in the short-medium term. In fact, I found that often this is not necessarily the case, or at least not the case a sufficiently high proportion of times that I became annoyed and started looking for a solution. Many times, concepts which were only briefly touched on in the text, which I thought of as minor footnotes at most, were in fact the key to solving a problem I was stuck on. Similarly, often I would know a concept was talked about in a book, but not remember the details (since it had been so long since I studied the material), so need to go in and refresh my memory on the relevant section, re-reading >10 min worth of material.
Thus I recommend using Anki or some other spaced-repetition program to remember the concepts you learn. My proficiency in subjects I learned after using Anki is tremendously higher than my proficiency in subjects I learned before using Anki, because it allows me to have an immediate and incredibly fast register of everything I’ve learned about in the relevant field for basically no work (<30 min a day).
Congratulations on the accomplishments! I did something similar a few years ago, using Spivak’s Calculus rather than Apostol’s (though I did consider using Apostol’s).
I used to think similarly to you, that I needn’t artificially reinforce concepts because those I actually need with be naturally reinforced in the wild, while those I don’t won’t, leaving me with only the skills I actually need.
However, this argument is subtly flawed. It assumes that concepts which are reinforced in the short-medium term are important, and concepts which are important are reinforced in the short-medium term. In fact, I found that often this is not necessarily the case, or at least not the case a sufficiently high proportion of times that I became annoyed and started looking for a solution. Many times, concepts which were only briefly touched on in the text, which I thought of as minor footnotes at most, were in fact the key to solving a problem I was stuck on. Similarly, often I would know a concept was talked about in a book, but not remember the details (since it had been so long since I studied the material), so need to go in and refresh my memory on the relevant section, re-reading >10 min worth of material.
Thus I recommend using Anki or some other spaced-repetition program to remember the concepts you learn. My proficiency in subjects I learned after using Anki is tremendously higher than my proficiency in subjects I learned before using Anki, because it allows me to have an immediate and incredibly fast register of everything I’ve learned about in the relevant field for basically no work (<30 min a day).
Thank you, that’s good to know. I’ll give it a download.
Cool! Let me know if it becomes valuable for you!