1. Let's play!; 2. Discovering and presenting mathematics; 3. Sets; 4. The integers and the fundamental theorem of arithmetic; 5. Functions; 6. Relations; 7. Cardinality; 8. The real numbers; 9. Probability and randomness; 10. Algebra and symmetry; 11. Projects; Appendix A. Solutions, answers, or hints to in-text exercises; Index; Bibilography.
With exercises and projects, Exploring Mathematics supports an active approach to the transition to upper-level theoretical math courses.
John Meier is Professor of Mathematics at Lafayette College, Pennsylvania, where he also served as Dean of the Curriculum. His research focuses on geometric group theory and involves algorithmic, combinatorial, geometric and topological issues that arise in the study of infinite groups. In addition to teaching awards from Cornell University, New York, and Lafayette College, Professor Meier is the proud recipient of the James Crawford Teaching Prize from the Eastern Pennsylvania and Delaware section of the Mathematical Association of America. Derek Smith is Associate Professor of Mathematics at Lafayette College, Pennsylvania. His research focuses on algebra, combinatorics, and geometry. He has taught a wide variety of undergraduate courses in mathematics and other subjects in both the United States and Europe. He is the recipient of multiple teaching awards at Lafayette, and his work has been supported by the Mathematical Association of America and the National Science Foundation. Professor Smith is a former editor of the problem section of Math Horizons.
'Wonderful list of topics, entertaining presentation, well-chosen
problems - this is the way I want my grandchildren to engage with
the beauty of mathematics. Too many students enter college with
tools from calculus but no concept of how mathematicians think, and
hardly any exposure to sets, logic, numbers, groups, or
probability. How can we expect them to decide about continuing with
mathematics, without a glimpse of the wonders ahead? A marvelous
world lies at your students' doorstep, and Exploring Mathematics
makes them players, not just observers, in that world. You'll want
all your math majors, and perhaps your computer science majors as
well, to have this experience.' Peter Winkler, Dartmouth College,
New Hampshire
'Meier and Smith have written a wonderful introduction to higher
mathematics, showing both the thrill of abstraction and the beauty
in discovering proofs.' Tom Garrity, Williams College,
Massachusetts
'I think that this is an excellent book. It is playful and serious
at the same time. The book takes the unusual step of explaining not
just mathematical topics but also what math is about and how to do
it. The book is full of interesting topics, exercises, and
examples. I think that this book could really fire up the
imagination of a student wanting to get a start in real
mathematics.' Richard Schwartz, Brown University, Rhode Island
'The biggest step in studying mathematics is learning to write
proofs. After calculus, students discover that truth is not a
matter of a calculation, but a careful argument, juggling concepts
within formal logic. Exploring Mathematics is a guide to this new
level. Rich with exercises and projects, the book provides a
well-grounded introduction to proof. 'Then the fun begins'. The
reader participates by doing what mathematicians do, experimenting,
formulating conjectures, exploring foundations for the basis of a
proof. This book engages the reader fully to reach the goal of
learning to fashion real mathematics. A valuable addition to every
undergraduate library.' John McCleary, Vassar College, New York
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