A Transition to Advanced Mathematics: A Survey Course

• Publisher:   Oxford University Press, USA
• Number Of Pages:   768
• Publication Date:   2009-07-27
• ISBN-10 / ASIN:   0195310764
• ISBN-13 / EAN:   9780195310764

A Transition to Advanced Mathematics: A Survey Course promotes the goals of a ``bridge'' course in mathematics, helping to lead students from courses in the calculus sequence (and other courses where they solve problems that involve mathematical calculations) to theoretical upper-level mathematics courses (where they will have to prove theorems and grapple with mathematical abstractions). The text simultaneously promotes the goals of a ``survey'' course, describing the intriguing questions and insights fundamental to many diverse areas of mathematics, including Logic, Abstract Algebra, Number Theory, Real Analysis, Statistics, Graph Theory, and Complex Analysis.
The main objective is "to bring about a deep change in the mathematical character of students -- how they think and their fundamental perspectives on the world of mathematics." This text promotes three major mathematical traits in a meaningful, transformative way: to develop an ability to communicate with precise language, to use mathematically sound reasoning, and to ask probing questions about mathematics. In short, we hope that working through A Transition to Advanced Mathematics encourages students to become mathematicians in the fullest sense of the word.
A Transition to Advanced Mathematics has a number of distinctive features that enable this transformational experience. Embedded Questions and Reading Questions illustrate and explain fundamental concepts, allowing students to test their understanding of ideas independent of the exercise sets. The text has extensive, diverse Exercises Sets; with an average of 70 exercises at the end of section, as well as almost 3,000 distinct exercises. In addition, every chapter includes a section that explores an application of the theoretical ideas being studied. We have also interwoven embedded reflections on the history, culture, and philosophy of mathematics throughout the text.

The authors of this book have two very different backgrounds in mathematics, and their teamwork allows for excellent discussions of all the topics that they cover. The section on complex analysis is particularly great as it is presented in a way that is certain to be accessible to any student that has completed the calculus sequence. Rather than quickly moving through various topics and bogging down the students with a long series of theorems, the authors present the complex analysis topics in a way that seems natural by concentrating on harmonic conjugates and the Cauchy-Reimann equations.
Many undergraduate students and instructors have trouble transitioning from the computational-based calculus to the proof-based topics in many higher-level math courses. This book is an excellent was to help students become familiar with the ideas behind proofs while simultaneously introducing them to several advanced topics.
As a former student of both of the professors and someone that has actually used a preliminary edition of the book in one of my undergraduate courses, I can safely assert that the authors certainly achieve their goal of creating a transition course that will help students understand the workings of advanced topics in mathematics.

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