This free, online text is designed to be used as 1-semester or 2-semester course for first-year mathematical sciences students.
The purpose of this text is to provide a first-year introduction to upper-level mathematics in the undergraduate curriculum. Not just calculus acknowledges that a single course on proof writing in the second year is too little, too late. Using topics that will be useful in later courses, this book accelerates the process of "mathematical maturity" through active reading, reasoning, problem solving, and writing.
This text can be used for a one-semester course, or for two half-credit courses spread over two semesters in the first year. This text is not intended to replace a formal introduction to proof writing. It is intended to widen student's perspectives beyond high school mathematics and introductory calculus courses.
The exposition is overtly designed for active reading. There are almost no examples, and almost no illustrating figures. Embedded reading exercises ("Checkpoints") require the reader to construct examples and make drawings. End-of-section problems ("Exercises") guide students in constructing short arguments to prove if-then propositions. An overall theme is that a few carefully chosen, powerful abstractions (such as: sets and functions, modular arithmetic, linear transformations of the plane, to name a few) form the basis for widely various applications.
David W. Lyons is a professor of mathematics at Lebanon Valley College in Annville, Pennsylvania, USA, where he has taught and conducted research since 2000. Lyons works in mathematical physics, leading a student-faculty research program in quantum information science since 2002. For more information, visit his academic website at the URL below.