Transition to Advanced Mathematics

<p>This unique and contemporary text not only offers an introduction to proofs with a view towards algebra and analysis a standard fare for a transition course but also presents practical skills for upper-level mathematics coursework and exposes undergraduate students to the context and culture of contemporary mathematics. </p><p>The authors implement the practice recommended by the Committee on the Undergraduate Program in Mathematics (CUPM) curriculum guide that a modern mathematics program should include cognitive goals and offer a broad perspective of the discipline.</p><p>Part I offers:</p><ol> <p> </p> <li>An introduction to logic and set theory.</li> <p> </p> <li>Proof methods as a vehicle leading to topics useful for analysis topology algebra and probability. </li> <p> </p> <li>Many illustrated examples often drawing on what students already know that minimize conversation about doing proofs.</li> <p> </p> <li>An appendix that provides an annotated rubric with feedback codes for assessing proof writing.</li> </ol><p>Part II presents the context and culture aspects of the transition experience including:</p><ol> <p> </p> <li>21st century mathematics including the current mathematical culture vocations and careers.</li> <p> </p> <li>History and philosophical issues in mathematics.</li> <p> </p> <li>Approaching reading and learning from journal articles and other primary sources.</li> <p> </p> <li>Mathematical writing and typesetting in LaTeX.</li> </ol><p>Together these Parts provide a complete introduction to modern mathematics both in content and practice. </p><p><strong>Table of Contents</strong></p><p> Part I - Introduction to Proofs</p><ol> <b> </b><p> </p> <li>Logic and Sets</li> <li><b>Arguments and Proofs</b></li> <li><b>Functions</b></li> <li><b>Properties of the Integers</b></li> <li><b>Counting and Combinatorial Arguments</b></li> <li> <strong>Relations<br><br>Part II - Culture History Reading and Writing</strong><br> </li> <li><b>Mathematical Culture Vocation and Careers</b></li> <li><b>History and Philosophy of Mathematics</b></li> <li><b>Reading and Researching Mathematics</b></li> <li><b>Writing and Presenting Mathematics</b></li> </ol><p><b>Appendix A. Rubric for Assessing Proofs</b></p><p><b>Appendix B. Index of Theorems and Definitions from Calculus and Linear Algebra</b></p><p><b>Bibliography</b></p><p>Index</p><p>Biographies</p><p>Danilo R. Diedrichs is an Associate Professor of Mathematics at Wheaton College in Illinois. Raised and educated in Switzerland he holds a PhD in applied mathematical and computational sciences from the University of Iowa as well as a master’s degree in civil engineering from the Ecole Polytechnique Fédérale in Lausanne Switzerland. His research interests are in dynamical systems modeling applied to biology ecology and epidemiology.</p><p><strong>Stephen Lovett </strong>is a Professor of Mathematics at Wheaton College in Illinois. He holds a PhD in representation theory from Northeastern University. His other books include <i>Abstract Algebra: Structures and Applications </i>(2015) <i>Differential Geometry of Curves and Surfaces</i> with Tom Banchoff (2016) and <i>Differential Geometry of Manifolds</i> (2019).</p>