The Companion Guide to The Mathematical Experience, Study Edition has been created as a teaching tool, not only for the teacher and the student, but also for those students who are potential teachers. Its major purpose is to enhance the value of The Mathematical Experience, Study Edition as a textbook for teachers and to provide content and method for prospective teachers. Thus, unlike instructional guides that are available to the adopting teacher only, this Companion is available to the student or the teacher who wants independently to develop further skills in teaching mathematics. An additional value is that it provides suggested topics to explore that are not in the text but that coordinate beautifully to the text. The inclusion of these topics makes The Companion Guide a flexible teaching tool, adaptable to a variety of courses and useable with many individual selections of other course materials. The Companion Guide is rich in suggestions for classroom discussion topics. Each is linked to a chapter of the textbook and to the central idea of learning how to think, talk, and write ABOUT mathematics while learning how to DO mathematics. It provides insights into the subtleties of mathematical concepts and warns of pitfalls where ambiguity and misunderstanding often arise. It is a wealth of experience with ideas that WORK, gained through live classroom interaction by the authors and shared in this book with the reader.