Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 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
- Solutions, Answers, or Hints to In-Text Exercises
- Bibliography
- Index
Preface
Published online by Cambridge University Press: 06 August 2018
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- 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
- Solutions, Answers, or Hints to In-Text Exercises
- Bibliography
- Index
Summary
Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof. This book introduces students to these facets of the field in a problem-focused setting. For over a decade, we and many others have used draft chapters of Exploring Mathematics as the primary text for Lafayette's Transition to Theoretical Mathematics course. Our collective experience shows that this approach assists students in their transition from primarily computational classes toward more advanced mathematics, and it encourages them to continue along this path by demonstrating that while mathematics can at times be challenging, it is also very enjoyable.
Here are some of the key features of Exploring Mathematics.
• The sections are short, and core topics are covered in chapters that present important material with minimal pre-requisites. This structure provides flexibility to the instructor in terms of pacing and coverage.
• Mathematical maturity requires both a facility with writing proofs and comfort with abstraction and creativity. We help students develop these abilities throughout the book, beginning with the initial chapters.
• A student does not learn mathematics by passive reading. It is through the creation of examples, questioning if results can be extended, and other such in-the-margin activities that a student learns the subject. We encourage this behavior by including frequent in-text exercises that serve not only to check understanding, but also to develop material.
• We construct many mathematical objects that are elementary in their definition and commonly referenced in upper-level classes. These are woven throughout the text, with related exercises providing numerous opportunities for independent investigations of their important properties.
• Each chapter concludes with a robust mixture of exercises ranging from the routine to rather challenging problems, and the book concludes with a collection of projects: guided explorations that students can work on individually or in groups.
These and other fundamental aspects of Exploring Mathematics are described in greater detail below, where we also indicate different ways an instructor can map out the material that can be covered in a single term.
An Active Approach
Our experience teaching courses that introduce students to mathematical proofs shows that the spirit of mathematics is effectively taught with a focus on problem-solving. It is in doing mathematics – by exploring definitions, forming conjectures, and working on the writing of proofs – that students begin to understand the discipline.
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- Chapter
- Information
- Exploring MathematicsAn Engaging Introduction to Proof, pp. xi - xviPublisher: Cambridge University PressPrint publication year: 2017
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