Is there a mathematical way of thinking? In my experience, there is. Some naturally think like one; some are well-trained to; and, some learn it the hard (and time-tested) way of trail-and-error. If you belong to the last category, here is a book that would grease your path and ease your journey — I am referring to Kevin Houston’s How to think like a mathematician.
Houston, in his preface to the book, describes the aim of the book as follows:
The aim of this book is to divulge the secrets of how a mathematician actually thinks. As I went through my mathematical career, there were many instance when I thought, ‘I wish someone had told me that earlier.’ This is a collection of such advice. Well, I hope it is more than such a collection. I wish to present an attitude — a way of thinking and doing mathematics that works — not just a collection of techniques (which I will present as well!)
I am glad to report that Houston succeeds admirably in his presentation of the attitude — at least as far as a non-mathematician like me (albeit with a keen interest in reading, understanding, using and to some extent even writing mathematics) can tell.
From the preface, I also understand that most of the methods described in the book had been tried and tested in the classrooms.
The book consists of six parts, namely,
 Study skills for mathematicians
 How to think logically
 Definitions, theorem and proofs
 Techniques of proof
 Mathematics that all good mathematicians need
 Closing remarks
Each part, in turn, consists of four to eight chapters; for example, the first section on study skills consists of one chapter on sets and functions (mainly to serve as a model mathematical text — to be used later for discussions), one chapter on mathematical reading, two on writing, and one on problem solving.
The chapters are full of exercises (both solved and unsolved); each of them also begin with a nice quote and end with a nice, short summary.
There is also a very useful appendix which gives the recipes for proving different types of mathematical statements.
Running into nearly 260 pages or so, and flowing nicely without any rough patches, here is a book that I would have recommended strongly — except for one small glitch; the book costs nearly nineteen pounds (nearly Rs. 1500/-), which makes it a book for libraries and not for individuals — I do not see many undergraduates who can afford such a costly book, especially when it is not a prescribed textbook, or, that they would buy even if they could afford it, considering that it is not even a hardbound edition.
I hope Cambridge brings out a low-priced edition soon. Till then, having exhausted my book grant for the year, I am going to be satisfied with a recommendation of the book to our library and with the downloaded near-final version of the book (available on Houston’s homepage as a pdf) for a personal copy.