I’ve done a substantial amount of independent learning in math, but almost none of it was through textbooks. (I am very bad at forcing myself to do textbook exercises.) Most of what I’ve learned outside of classrooms has been from a combination of Wikipedia, reading math blog posts, and writing math blog posts. My main substitute for doing exercises is trying to work through the details of a result I find interesting on my blog; I try to prove everything that leads to or fits in the context of the result along the way, and those are my exercises. (Edit: my secondary substitute is solving problems on sites like math.stackexchange.)
It would be very helpful if you were more specific. What’s an example of an exercise in Spivak that took you a long time, and what did you do during that time?
There wasn’t a single problem in particular I found took me a long time. It was more… when I finished a set of exercises, I looked at the clock and realized a couple of hours have gone by, and that seemed way too long (though it really wasn’t).
For example, the first chapter deals with properties of numbers (multiplicative/additive inverses and identities, associative laws, etc.) and using these properties in proofs. Some of the exercises, say, #4, are split up into several other exercises (i, ii, iii, iv, etc.). “Prove: if a > 1, then a^2 > a”—Which seemed fairly trivial to me at first, but proving them using only the properties of numbers/inequalities provided made the task more difficult.
During the time it took for me to solve them, I just… well… sat down and solved them. Played around with what I could and couldn’t do, referred to some proofs he mentioned earlier, etc. :)
I’ve done a substantial amount of independent learning in math, but almost none of it was through textbooks. (I am very bad at forcing myself to do textbook exercises.) Most of what I’ve learned outside of classrooms has been from a combination of Wikipedia, reading math blog posts, and writing math blog posts. My main substitute for doing exercises is trying to work through the details of a result I find interesting on my blog; I try to prove everything that leads to or fits in the context of the result along the way, and those are my exercises. (Edit: my secondary substitute is solving problems on sites like math.stackexchange.)
It would be very helpful if you were more specific. What’s an example of an exercise in Spivak that took you a long time, and what did you do during that time?
There wasn’t a single problem in particular I found took me a long time. It was more… when I finished a set of exercises, I looked at the clock and realized a couple of hours have gone by, and that seemed way too long (though it really wasn’t).
For example, the first chapter deals with properties of numbers (multiplicative/additive inverses and identities, associative laws, etc.) and using these properties in proofs. Some of the exercises, say, #4, are split up into several other exercises (i, ii, iii, iv, etc.). “Prove: if a > 1, then a^2 > a”—Which seemed fairly trivial to me at first, but proving them using only the properties of numbers/inequalities provided made the task more difficult.
During the time it took for me to solve them, I just… well… sat down and solved them. Played around with what I could and couldn’t do, referred to some proofs he mentioned earlier, etc. :)
That sounds normal. Spivak isn’t an easy book, from what I’ve been told.