An Introduction to Writing Mathematical Proofs,

Edition 1 Shifting Gears from Calculus to Advanced Mathematics

Editors: By Thomas Bieske

Publication Date: 13 Jan 2026

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An Introduction to Writing Mathematical Proofs: Shifting Gears from Calculus to Advanced Mathematics addresses a critical gap in mathematics education, particularly for students transitioning from calculus to more advanced coursework. It provides a structured and supportive approach, guiding students through the intricacies of writing proofs while building a solid foundation in essential mathematical concepts. Sections introduce elementary proof methods, beginning with fundamental topics such as sets and mathematical logic, systematically develop the properties of real numbers and geometry from a proof-writing perspective, and delve into advanced proof methods, introducing quantifiers and techniques such as proof by induction, counterexamples, contraposition, and contradiction.

Finally, the book applies these techniques to a variety of mathematical topics, including functions, equivalence relations, countability, and a variety of algebraic activities, allowing students to synthesize their learning in meaningful ways. It not only equips students with essential proof-writing skills but also fosters a deeper understanding of mathematical reasoning. Each chapter features clearly defined objectives, fully worked examples, and a diverse array of exercises designed to encourage exploration and independent learning. Supplemented by an Instructors' Resources guide hosted online, this text is an invaluable companion for undergraduate students eager to master the art of writing mathematical proofs.

Key Features

Introduces foundational topics in elementary proof methods, including sets, mathematical logic, properties of real numbers, and geometry to establish a strong basis for proof writing
Helps undergraduate students develop or enhance their proof-writing abilities, particularly those in STEM fields with a background in Calculus I.
Fills a critical gap in mathematics education by providing structured guidance for students transitioning to higher-level proof-oriented mathematics
Offers a wealth of resources, including clearly defined objectives, fully worked examples, and diverse exercises to encourage exploration and independent learning
Supplemented by an Instructor Resource Guide that includes writing prompts, group projects, and group brainstorming activities

About the author

By Thomas Bieske, Chair of the Undergraduate Committee-Upper Level, Department of Mathematics and Statistics, University of South Florida, Tampa., USA

1. Introduction

Section I: Elementary Proof Methods: Our First Bicycle
2. Sets and Notation - Introduction to basic set theory
3. Mathematical Logic - Basic logic needed to be able to be able to write proofs.
4. Properties of Real Numbers - Systematically builds up the properties of real numbers
5. Geometry Revisited - Approaches topics from high school geometry from the point of view of proof writing.

Section II: Advanced Proof Methods: Bicycles with Multiple Gears
6. Quantifiers and Induction - Introduces quantifiers and the technique of proof by induction
7. The Three C's: Counterexamples, Contraposition, and Contradiction - Introduces these indirect proof methods

Section III: Using Our Techniques: A Mathematical Tour de France
8. Fun with Functions and Relations - An Exploration and Opportunities for writing proofs involving functions and relations
9. An Amalgam of Algebraic Activities - Opportunities to write proofs for algebraic topics
10. Appendix
11. Index

ISBN: 9780443439247

Page Count: 122

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Access to teacher/student resources is available to registered users with approved review copies or confirmed adoptions. To review this material, please request a review copy.