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. T

he 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.

**Answer**

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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