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I have been working through Kenneth P. Bogart's "Combinatorics Through Guided Discovery". You can download it from this page: http://www.math.dartmouth.edu/news-resources/electronic/kpbogart/

I've found that it is a great way to learn and makes me think about the concepts as if I were discovering them. I think that a lot of people will find benefit in working through such a book.

I've looked for books in the same spirit as this one: learning through guided discovery, but my searches haven't been fruitful.

Does anyone know of any such books?

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This guided discovery approach goes by other names, as well. One such name is "Inquiry Based Learning" or IBL. A list of guided discovery problems is often referred to as an "IBL script". Many such scripts are available from the Journal of Inquiry Based Learning in Mathematics (JIBLM): http://www.jiblm.org/

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  • $\begingroup$ @Jon Bannon: The JIBLM site is great and seems full of great books (I have yet to look through all of them). I'd up vote if I could (my reputation is <15). Thanks! $\endgroup$ Jan 27, 2013 at 4:25
  • $\begingroup$ @Théophile: I'm happy to pass this along. $\endgroup$
    – Jon Bannon
    Jan 27, 2013 at 13:39
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You may be interested in learning about the Moore Method. The idea is to "encourages students to solve problems using their own skills of critical analysis and creativity" without relying on textbooks. HERE you can find some references.

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I have not read (or, in this case, worked) through the book, but Jeffrey Strom's ``Modern Classical Homotopy Theory" guides the reader through the proofs of all the theorems stated in the book (as opposed to proving them himself). To my very limited knowledge, this is the first "IBL-type" book in algebraic topology.

This is the book:

http://books.google.com/books/about/Modern_Classical_Homotopy_Theory.html?id=ruhFJoMIIPoC

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I'm impressed with two books by Dr R. P. Burn that seem to be in the spirit of your question:-

  1. Groups: a path to geometry, CUP, 1985, 0-521-30037-1
  2. A pathway to number theory, CUP, 2nd ed., 1997, 978-0-521-57540-9

Each consists of an ordered sequence of problems (answers provided):-

... to enable students to participate in the formulation of central mathematical ideas before a formal treatment (which, suitably introduced, they may well be able to provide themselves)

Source: a preface to A pathway to number theory)

They are aimed at advanced high school, or early undergraduate level students. The sequence starts by getting the reader to initially explore special cases and then work towards a generalisation, usually a theorem. The books include references to selected standard texts that are recommended to be read concurrently.

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  • $\begingroup$ Burn also has a book on introductory real analysis, which is quite good. $\endgroup$
    – 5space
    Aug 16, 2014 at 9:06
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My favourite is Alexandre Kirillov's "Elements of the Theory of Representations" Grundlehren der Mathematischen Wissenschaften, Springer, vol 220. A lot of representation theory is worked out through examples and exercises.

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Vakil's Rising Sea as a first introduction to modern algebraic geometry!

There is also quite a psychological element in his writing. Often he gives really long hints to his exercises. If one compares such hints with classical texts, then one often finds that he does not give fewer details than classical texts! Indeed, often these hints were actual proofs of theorems in a previous version of his notes which he turned to hints of an exercise.

The psychological effect is the following. While e.g. Hartshorne write "We have x, and thus y", Vakil writes "Convince yourself of x. Show that this implies y.", so in the first instance I would get frustrated as to why I can't understand a seemingly simple proof but reading Vakil I would be proud to have solved another exercise and the step would better stick to my memories.

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Linear Algebra Problem Book by P.R. Halmos is written very much in this spirit: learning through guided discovery. I use it for my "Advanced Investigations in Linear Algebra" course.

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Similar to his "Linear Algebra Problem Book", Halmos also wrote "A Hilbert Space Problem Book". I have only skimmed it but it seems as good as LAPB which I remember liking a lot.

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The book Abel's Theorem in Problems and Solutions by Alekseev & Arnold is a great one to learn about group theory and complex analysis (see excerpts here)

Also, have a look at the following related MO questions: 12709, 28158 and 56314.

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  • $\begingroup$ @Thomas Sauvaget: Thanks! Question 12709 is very similar to mine and more precise. I didn't see it before creating this one... Is there any way to merge them? $\endgroup$ Jan 27, 2013 at 3:17
  • $\begingroup$ The book you cited and the answers you got are complementary to 12709, so there's nothing to worry about, I think. $\endgroup$ Jan 27, 2013 at 8:40
  • $\begingroup$ Nice book, I love it. $\endgroup$
    – bzm3r
    Nov 22, 2015 at 5:23
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You may find this one interesting: Number Theory Through Inquiry (MAA textbooks). I have used it three times. First time, which I strictly followed the method, we just coverd the first four chapters. Second time, I have relaxed myself a bit and we covered the first six chapters. Last time (current term), I have used all the teaching methods I know (including modified Moore method), we are nearly covering all chapters!

You may also find this paper interesting: "http://www.tandfonline.com/doi/abs/10.1080/10511970.2011.639337#preview">Moore and Less" (PRIMUS,22(7):509-524, 2012) where I told the story of using a very modified Moore method in a Multivariable Calculus Course.

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  • $\begingroup$ Your paper sounds interesting, but unfortunately, for me at least, it seems to be behind a paywall. $\endgroup$
    – J W
    May 5, 2013 at 5:10
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    $\begingroup$ Send an e-mail to my gmail: asghari.amir, I'll send you a copy via PRIMUS. $\endgroup$ May 5, 2013 at 6:36
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I've just put up such a text for an Introduction to Proofs course, here. It is Free, including LaTeX source. (I've only taught out of it one time so no doubt there are typos, places that could use refinement, etc.)

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