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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?

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This is really a matter of debate: it varies according to both learners and topic, as well as the level of study, and there are proponents/opponents of both approaches. See for example: the Moore-Method of teaching (and/or writing text-books), and its applications and proponents. –  amWhy Jan 9 '13 at 23:22
If you write your own reviews for such a book, the above problems would be solved. –  Makoto Kato Jan 9 '13 at 23:23
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2 Answers

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.

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I didn't know the Moore Method, but after a quick tour on Wikipedia, I would like if someone here has ever try this method ? I am very interested at the educational result on someone. –  Alan Simonin Jan 9 '13 at 23:33
Moore used it extensively with his students. Here is a list of his advisees: genealogy.math.ndsu.nodak.edu/id.php?id=286 For what it's worth, I wrote up a paper recommending prospective mathematics teachers experience at least one non-lecture course at some point in their education. In particular, I outlined a course on topology using the Kuratowski Closure Axioms, where students began by considering what "close" should mean in an abstract setting. The class would then be taught with the Moore Method, with the main topics being: closed (and open) sets, connectedness, and continuity. –  Benjamin Dickman Jan 9 '13 at 23:38
Thanks for the answers ! I find this method wonderful, because most of regular students just "copy" methods for proofs and apply them to the exercises. It seems that the method develop a good intuition and maybe a better taste of what research looks like. On the other hand, you can't cover as much themes as in a regular way of teaching and some theorems could be very difficult to proove without having seen the proof once before –  Alan Simonin Jan 9 '13 at 23:49
@AndresCaicedo Thank you for your link, I looked the video and found them very interesting ! –  Alan Simonin Jan 10 '13 at 0:38
@B.D I read your paper and found it very interesting ! I hope this method will be used more and more –  Alan Simonin Jan 13 '13 at 13:57
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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.

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