I enjoyed the book very much because of its thorough account of the major and eccentric people who worked to prove or disprove the Poincare Conjecture. I was either working on my undergraduate or in grad school when the Perelman papers were published. I remember hearing about it from faculty and fellow graduate students but that is not why I am mentioning it here.
The descriptions of the amount of (sometimes "wasted") effort on solving the conjecture illustrate the difficulty of the problem and the tenacity needed to solve ANY problem. My students often seem to think they are either good or bad at math. Either opinion is reason to give up if the solution doesn't come quickly. I don't know how I will do this but I want to find a couple excerpts about how either people kept returning to the problem or were very occupied with it. I think a major point to discuss would be that a problem has an essential question to be answered (which will hopefully tell us something interesting and novel about the world) which is different from what is learned by working out the technical details. I would hope this would turn into a discussion about why we learn math which hopefully is because we have insatiable curiosities.
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