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I have been working through Kenneth P. Bogart's "Combinatorics Through Guided Discovery". You can download it from this page:

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 should be Community Wiki, I think (edit the question and tick the "Community Wiki" box). – Daniel Moskovich Jan 23 '13 at 6:29
@Daniel Moskovich: done. – Théophile Cantelobre Jan 23 '13 at 6:45
The following is related: Theorems as Problems. – 5space Aug 4 '14 at 1:41

10 Answers 10

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):

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@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! – Théophile Cantelobre Jan 27 '13 at 4:25
@Théophile: I'm happy to pass this along. – Jon Bannon Jan 27 '13 at 13:39

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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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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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:

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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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Burn also has a book on introductory real analysis, which is quite good. – 5space Aug 16 '14 at 9:06

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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This is not to criticise you as I know it is not very well known, which is why I say it: if you think a question should be in CW mode, you could on the one hand give your answer right away in CW mode. And, this is the main point, if you do not or there are already other answers in non-CW mode, you should better flag for moderator attention with request to turn into CW (as opposed to asking the OP since they can just CW the question, which then leaves your and other answers in non-CW mode) [I now already flagged for mods so no action is required, this is just for info.] – user9072 Jan 23 '13 at 10:51

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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@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? – Théophile Cantelobre Jan 27 '13 at 3:17
The book you cited and the answers you got are complementary to 12709, so there's nothing to worry about, I think. – Thomas Sauvaget Jan 27 '13 at 8:40
Nice book, I love it. – user89 Nov 22 '15 at 5:23

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: "">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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Your paper sounds interesting, but unfortunately, for me at least, it seems to be behind a paywall. – J W May 5 '13 at 5:10
Send an e-mail to my gmail: asghari.amir, I'll send you a copy via PRIMUS. – Amir Asghari May 5 '13 at 6:36
Thanks! Much appreciated. – J W May 5 '13 at 13:54

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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