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K. P. Bogart wrote Combinatorics through Guided Discovery, available freely online. In the preface, he writes (emphasis mine):

The point of learning from this book is that you are learning how to discover ideas and methods for yourself, not that you are learning to apply methods that someone else has told you about. The problems in this book are designed to lead you to discover for yourself and prove for yourself the main ideas of combinatorial mathematics. There is considerable evidence that this leads to deeper learning and more understanding.

Can you recommend other books that are similar? Note that "guided discovery" can take a few different forms.

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  • $\begingroup$ Shouldn't this be a CW? $\endgroup$ – Dair Jan 15 '15 at 5:15
  • $\begingroup$ @Bair Yes, it should be -- could you remind me how I can do that? $\endgroup$ – user89 Jan 15 '15 at 5:16
  • $\begingroup$ There should be a check box if you go to edit the question. $\endgroup$ – epimorphic Jan 15 '15 at 17:52
  • $\begingroup$ Could you be slightly more specific? Are we talking about undergraduate or graduate level mathematics? Or at any level? Also, do you want to know about books for certain subjects in particular? $\endgroup$ – Daniel W. Farlow Jan 18 '15 at 23:27
  • $\begingroup$ @induktio Unless specified, there are no specifications. $\endgroup$ – user89 Jan 18 '15 at 23:28

10 Answers 10

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Linear Algebra Problem Book by Halmos. From the description:

Can one learn linear algebra solely by solving problems? Paul Halmos thinks so, and you will too once you read this book. The Linear Algebra Problem Book is an ideal text for a course in linear algebra. It takes the student step by step from the basic axioms of a field through the notion of vector spaces, on to advanced concepts such as inner product spaces and normality. All of this occurs by way of a series of 164 problems, each with hints and, at the back of the book, full solutions.

Distilling Ideas: An Introduction to Mathematical Thinking by Katz and Starbird.

This book gives an inquiry based learning approach to some topics in graph theory, group theory, and calculus.

Number Theory Through Inquiry by Marshall, Odell, and Starbird. From the description:

Number Theory Through Inquiry; is an innovative textbook that leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. One goal is to help students develop mathematical thinking skills, particularly, theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for an independent study experience, or for a course designed as an introduction to abstract mathematics.

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Here are two candidates providing completely different methods of guided discovery. The first one with focus on visual aha experience:

Visual Complex Analysis by T. Needham is a guided tour through Complex Analysis with plenty of illuminating pictures providing additional insight (and additional aesthetic pleasure).

From the preface: In the hope of making the book fun to read, I have attempted to write as though I were explaining the ideas directly to a friend. Correspondingly, I have tried to make you, the reader, into an active participant in developing the ideas.

While Needham uses the visual senses as additional support to ease reading and enhance understanding, David Bressoud uses time or more precisely the historical development of mathematical ideas as guiding principle.

In A Radical approach to Real Analysis the course does not follow the traditional development, namely starting with a discussion of properties of real numbers, then moving on to continuity, then differentiability and so forth. He instead takes the reader along the often devious chronological pathes of development and forces him this way to think about concepts and so to grasp essential ideas.

From the preface: ... the first part of this book ... starts with infinite series ... illustrating the great successes that led the early pioneers onward, as well as the obstacles that stymied even such luminaries as Euler and Lagrange. There is an intentional emphasis on the mistakes that have been made. These highlight difficult conceptual points. ... The student needs time with them. The highly refined proofs that we know today leave the mistaken impression that the road of discovery in mathematics is straight and sure. It is not. Experimentation and misunderstandng have been essential components in the growth of mathematics.

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  • $\begingroup$ Good job on showing that "guided discovery" can take a few different forms. $\endgroup$ – user89 Jan 18 '15 at 23:16
  • $\begingroup$ @user89: Thanks and good to see that we agree. $\endgroup$ – Markus Scheuer Jan 18 '15 at 23:26
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    $\begingroup$ the best is the second: explaining the motivations exactly is superb. the history of mathematics is as important as mathematics itself. $\endgroup$ – user499752 Jan 15 '18 at 20:35
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My answer is quite philosophical and the books concerned with guided discovery, which I read, are less concrete than proposed by the others:

  • George Polya, “Mathematical discovery: on understanding, learning and teaching”
  • Henri Poincaré, “Mathematical Creation”.
  • Imre Lakatos, “Proofs and refutations. The Logic of Matematical Discovery
  • Jacques Hadamard, “An essay on The Psychology of Invention in the Mathematical Field”
  • Louis Mordell, “Reflections of a mathematician”
  • Karl Popper, “The Logic of Scientific Discovery”
  • Simon Singh, “Fermat's last theorem” (how they did it)

I hope that the titles of the books are descriptive. I wish to present short abstracts for these books, but it is hard to me. These guys are famous, and I can say to a seeker: Cast a glance at the book. It you’ll found it good for you it’ll be OK. If not – this also be OK, there are a lot of other books in the world. :-)

Also some useful advices can be found from this page by Terence Tao, who is a Fields Prize Winner.

Instead of the abstracts I can share with a reader my general point of view on the usage of such books. Thus a reader not interested in ideology can finish read my answer here.

I believe that principles of scientific discovery are objective, because a method which regularly leads to a success, which was successful not only for a first discovery, but which will lead us to the future discoveries, have to be based on objective, universal foundations, such as gravitation laws or mathematical arguments. From this point of view history of science is a concrete illustration of general principles. Therefore I believe that a right way for a scientist is not to deal independently, but to deal good. Leonardo da Vinci said: “There are three classes of people: those who see, those who see when they are shown, those who do not see”. So I want to believe that an urge to independent thinking results from poverty, therefore I wish that a seeker could say, like Nietzsche's Zarathustra: “I am not poor enough for that”. :-) I think that a right way is a way of masters, because both masterity and masters are winners in concurrent struggle of alternatives, they endure selection and trials (by life), proved its effectiveness, success. Therefore way of masters is in some sense a natural way. Life stands problems before us (due to nature of the world) and we strive to solve them (due to our nature). Thus a masters way is about deep acquirement and use of masters’ (effective) methods (strategies, doctrines, theories, schemes, modes of thought and act, moods, at last), form and development of intuition, improvement of masterity. Then great masters books stands as lighthouses along a way of a seeker, who wish to follow masters way, do as they do.

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'A Pathway into Number Theory' and 'Groups: A Pathway into Geometry' both by R.P. Burn

Introduces both topics solely through a sequence of questions and exercises.

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There is the voluminous (835 big format pages) treatise Modern Classical Homotopy Theory (Graduate Studies in Mathematics) by Jeff Strom. It teaches the reader quite a bit of homotopy theory almost exclusively through guided exercises.

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John Greever’s Theory and Examples of Point-Set Topology probably qualifies. It’s a topology text designed to be used with some version of the Moore method: some of the harder theorems and examples are treated in detail, but for the most part the book is an organized presentation of definitions and statements of theorems that students are to prove on their own. I had the pleasure of first encountering topology in Greever’s course in $1966$, when the book was still a set of dittoed notes.

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Classic Set Theory: for Guided Independent Study by Derek Goldrei is supposed to be a "guided discovery" book, but I think it reads pretty much like any standard textbook.

Might be worth a try though for those interested in set theory.

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A. Gardiner, Discovering Mathematics

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