Mathematics is not a spectator sport: successful students of mathematics grapple with ideas for themselves. Distilling Ideas presents a carefully designed sequence of exercises and theorem statements that challenge students to create proofs and concepts. As students meet these challenges, they discover strategies of proofs and strategies of thinking beyond mathematics. In order words, Distilling Ideas helps its users to develop the skills, attitudes, and habits of mind of a mathematician and to enjoy the process of distilling and exploring ideas. Distilling Ideas is an ideal textbook for a first proof-based course. The text engages the range of students' preferences and aesthetics through a corresponding variety of interesting mathematical content from graphs, groups, and epsilon-delta calculus. Each topic is accessible to users without a background in abstract mathematics because the concepts arise from asking questions about everyday experience. All the common proof structures emerge as natural solutions to authentic needs. Distilling Ideas or any subset of its chapters is an ideal resource either for an organized Inquiry Based Learning course or for individual study. A student response to Distilling Ideas: "I feel that I have grown more as a mathematician in this class than in all the other classes I've ever taken throughout my academic life."
Daugiau informacijos
A guide to mathematical thinking for undergraduates and lecturers which uses exercises and theorems to develop mathematical creativity.
| Preface |
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1.1 Proof and Mathematical Inquiry |
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2.1 The Konigsberg Bridge Problem |
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2.9 Completing the Walk around Graph Theory |
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3.1 Examples Lead to Concepts |
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3.2 Clock-Inspired Groups |
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3.3 Symmetry Groups of Regular Polygons |
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3.4 Subgroups, Generators, and Cyclic Groups |
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3.5 Sizes of Subgroups and Orders of Elements |
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3.9 Normal Subgroups and Quotient Groups |
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3.12 The Man Behind the Curtain |
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4.7 Speedometer Movie and Position |
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4.8 Applications of the Definite Integral |
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4.9 Fundamental Theorem of Calculus |
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141 | (4) |
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4.10 From Vague to Precise |
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145 | (4) |
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149 | (4) |
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149 | (4) |
| Annotated Index |
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153 | (12) |
| List of Symbols |
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165 | (4) |
| About the Authors |
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Brian Katz is an Assistant Professor of Mathematics at Augustana College in Rock Island, Illinois. He received his BA from Williams College in 2003 with majors in mathematics, music and chemistry, and his PhD from the University of Texas, Austin in 2011, concentrating on algebraic geometry. While at the University of Texas, Austin, Brian received the Frank Gerth III Graduate Excellence Award and the Frank Gerth III Graduate Teaching Excellence Award from the Department of Mathematics. Brian is a Project NExT Fellow, supported by Harry Lucas, Jr and the Educational Advancement Foundation. Michael Starbird is Professor of Mathematics and a University Distinguished Teaching Professor at the University of Texas, Austin. He received his BA degree from Ponoma College and his PhD in mathematics from the University of Wisconsin, Madison. He has held visiting positions at the Institute for Advanced Study in Princeton, New Jersey, and at the Jet Propulsion Laboratory in Pasadena, California. He served as Associate Dean in the College of Natural Sciences at the University of Texas from 1989 to 1997. Starbird's mathematical research is in the field of topology. He has served as a member-at-large of the Council of the American Mathematical Society and on the national education committees of both the American Mathematical Society and the Mathematical Association of America.
