This book, which is based on Pólya's method of problem solving, aids students in their transition
from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal
of guidance on how to approach definitions, examples, and theorems in mathematics.
It ends by providing projects for independent study.
Students will follow Pólya's four-step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.
New features of the Second Edition include the addition of formal statements of definitions at the end of each chapter; a new chapter featuring the Cantor-Schröder-Bernstein theorem with a spotlight on the continuum hypothesis; over 200 new problems; two new student projects; and more. An electronic solutions manual to selected problems is available on this website in Resources.
From the reviews of the First Edition:
"The book Reading, Writing, and Proving: A Closer Look at Mathematics (abbreviated RWP in this review) provides a fresh, interesting, and readable approach to the often-dreaded "Introduction to Proof" class. ... RWP emphasizes Pólya's four-part framework for problem solving (from his book How to Solve It). ... Beautiful examples of proofs containing the right mix of formalization and language are provided throughout the book. ... RWP puts a great deal of effort into casting the skill of constructing proofs as a problem-solving process. ... RWP contains more than enough material for a one-semester course, and is designed to give the instructor wide leeway in choosing topics to emphasize. ... This book has a rich selection of problems for the student to ponder, in addition to "exercises" that come with hints or complete solutions. ... I was charmed by this book and found it quite enticing."
-- Marcia G. Fung for MAA Reviews
"The book is well written and should be easy for a first- or second-year college mathematics student to read. There are many "tips" offered throughout, along with many examples and exercises of appropriate type and difficulty. ... A book worthy of serious consideration for courses whose goal is to prepare students for upper-division mathematics courses. Summing Up: Highly recommended."
-- J. R. Burke, Gonzaga University for CHOICE Reviews