A Transition to Advanced Mathematics: A Survey Course promotes the goals of a "bridge'" course in mathematics, helping students transition from courses in the calculus sequence (and other problem solving courses that involve mathematical calculations) to theoretical upper-level mathematics courses (that prove theorems and grapple with mathematical abstractions). Simultaneously, the text promotes the goals of a "survey'" course, describing intriguing questions and insights fundamental to diverse areas of mathematics, including Logic, Abstract Algebra, Number Theory, Real Analysis, Statistics, Graph Theory, and Complex Analysis.
The text's main objective is to provide a deep change in the mathematical character of students—in their thought process and in their fundamental perspective on the mathematical world. We believe 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.
The text has a number of distinctive features that enable this transformational experience. Embedded Questions within each section and Reading Questions at each section’s end illustrate and explain concepts, allowing students to test their understanding independent of the exercises. Extensive, diverse exercise sets contain an average of 70 exercises at the end of each section; in total, the text has almost 3000 distinct exercises. In addition, every chapter includes a section that explores an application of the theoretical ideas being studied. Finally, embedded reflections on the history, culture, and philosophy of mathematics are interwoven throughout the text.
Bridge. Survey. Challenging. Encouraging. Reflective. Practical. Transformational. A Transition to Advanced Mathematics: A Survey Course will make a positive difference in the learning and development of your mathematics majors.