An Introduction to Writing Mathematical Proofs: Shifting Gears from Calculus to Advanced Mathematics - Softcover

Bieske

Softcover

ISBN 10: 0443439249 ISBN 13: 9780443439247

Publisher: Academic Press, 2026

Synopsis

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.

• 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 this title

About the Author

Professor Thomas Bieske earned his PhD from the University of Pittsburgh, United States, in 1999. His research concerns partial differential equations and analysis in metric spaces, with a focus on sub-Riemannian spaces. Professor Bieske is currently serving as the Department of Mathematics and Statistics Chair of the Undergraduate Committee-Upper Level, focusing on the performance of mathematics and statistics majors in upper-level courses, at the University of South Florida, Tampa, United States.

From the Back Cover

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 material. Recognizing the challenges many students face in proof-writing and mathematical logic, the book provides a structured and supportive approach, guiding students through the intricacies of writing proofs while building a solid foundation in essential mathematical concepts.

The book is organized into three comprehensive sections. The first section introduces elementary proof methods, from sets and mathematical logic to systematically developing the properties of real numbers and geometry from a proof-writing perspective. The second section delves into advanced proof methods, introducing quantifiers and techniques, including proof by induction, counterexamples, contraposition, and contradiction. Finally, the third section applies these techniques to a variety of mathematical topics, such as functions, absolute value, and modular arithmetic, allowing students to synthesize their learning.

An Introduction to Writing Mathematical Proofs helps in 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.

Edition details

• Publisher: Academic Press
• Publication date: 2026
• Language: English
• ISBN 10: 0443439249
• ISBN 13: 9780443439247
• Binding: Paperback
• Number of pages: 122