How does one best balance learning from a “problem based book” with supplementary material?

Question

We all know that when learning math, one has to do more than just simply read - one must try to solve problems and work actively with the material.

Many books try to force the reader to participate in the learning experience by actually consisting of no proofs (or close to none), with some guidance to the big ideas. This is clearly a valuable approach, but it can sometimes come at the price of:

1. Making (quick) reviews somewhat hard, since you have to rework out all the details, and it is easy to think that one has reviewed something in full detail, when one really hasn't.
2. One might progress a bit slower than usual, which is obviously not ideal. One might argue that the slower pace is compensated by actually learning the material deeper the first time, but how does this compare to reviewing the material several times (which for me, can be done more easily by a "traditional" book)?

My main question(s) are the following:

What should one think about when trying to learn a subject through a problem based book? Should one try to supplement it with something not containing only problems, so that one maybe get a deeper exposition from someone who has already digested the material?

Answer 1

If your goal is a quick review, then the sort of book you are discussing is likely not the right choice.

Yes, there is a clear tradeoff between speed and, well, something else. Without obsessing over the precision of the language I'm using, I'll just call this something else experience actually doing mathematics.

With regard to your main questions, I am afraid there can be no one answer. As I said above, if time is of the essence you might consider using a different book. At the opposite end of the spectrum, you have the teaching style of R.L. Moore (look up articles on "The Moore Method") in which students who were discovered as having sought supplementary material were banished from his class. If you are asking my own personal opinion, I'd say the case here is as it is with most involving some sort of spectrum of teaching approaches: the ideal choice lies somewhere in the middle. Ultimately, though, there is no definitive answer.

In any event, here are some references related to the Moore Method in case it interests you:

1 The Origin and Early Impact of the Moore Method. David E. Zitarelli. The American Mathematical Monthly, Vol. 111, No. 6 (Jun. - Jul., 2004), pp. 465-486. http://www.jstor.org/stable/4145066.

[2] The Moore Method. F. Burton Jones. The American Mathematical Monthly, Vol. 84, No. 4 (Apr., 1977), pp. 273-278. http://www.jstor.org/stable/2318868.

[3] Student Oriented Teaching: The Moore Method. Lucille S. Whyburn. The American Mathematical Monthly, Vol. 77, No. 4 (Apr., 1970), pp. 351-359. http://www.jstor.org/stable/2316141.

[4] A Modified Moore Method for Teaching Undergraduate Mathematics. David W. Cohen. The American Mathematical Monthly, Vol. 89, No. 7 (Aug. - Sep., 1982), pp. 473-474, 487-490. http://www.jstor.org/stable/2321385.

Edit: Here are a bunch more references about the Moore Method.

Answer 2

If you can initially work most of the problems and after a chapter or two you can work the problems you could not do initially then you might not need help. On the other hand, if a majority of the problems are unworkable then you need outside assistance. Perhaps a person who knows the material is better because they can point out the exact points that are giving you trouble. Otherwise find a book that has proofs in some detail. Also you may want to ask yourself if you know the prerequisites for the material as well as you should.