Distilling Ideas : An Introduction to Mathematical Thinking.

Cover -- copyright page -- title page -- Contents -- Preface -- Introduction -- Proof and Mathematical Inquiry -- Graphs -- The Königsberg Bridge Problem -- Connections -- Taking a Walk -- Trees -- Planarity -- Euler Characteristic -- Symmetries -- Colorability -- Completing the Walk around Graph Theory -- Groups -- Examples Lead to Concepts -- Clock-Inspired Groups -- Symmetry Groups of Regular Polygons -- Subgroups, Generators, and Cyclic Groups -- Sizes of Subgroups and Orders of Elements -- Products of Groups -- Symmetric Groups -- Maps between Groups -- Normal Subgroups and Quotient Groups -- More Examples* -- Groups in Action* -- The Man Behind the Curtain -- Calculus -- Perfect Picture -- Convergence -- Existence of Limits -- Continuity -- Zeno's Paradox™ -- Derivatives -- Speedometer Movie and Position -- Applications of the Definite Integral -- Fundamental Theorem of Calculus -- From Vague to Precise -- Conclusion -- Distilling Ideas -- Annotated Index -- List of Symbols -- About the Authors -- Back cover.

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 other 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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.