Counter-Examples in Calculus

Welcome to the interactive space for people interested in counter-examples in calculus. 

Ako Aotearoa is supporting the distribution of complimentary copies of the publication: Counter-Examples in Calculus by Associate Professor Sergiy Klymchuk, from AUT University.  We are distributing these books to all secondary schools and university mathematics departments in New Zealand. If you have not yet received a copy, please contact Sergiy direct by clicking on his name in the right hand side bar.

The distribution of this book is one of Ako Aotearoa’s initiatives to support the transition of learners from secondary to tertiary education. You can read more about Sergiy's use of counter-examples in a recent publication under our Good Practice Publication Grants Scheme illustrating the use of counter-examples with undergraduate learners - Using Counter-Examples to Enhance Learners’ Understanding of Undergraduate Mathematics.

This book is a supplementary resource intended to enhance the teaching and learning of a first-year university Calculus course. It can also be used in upper secondary school. It consists of carefully constructed incorrect mathematical statements that require students to create counter-examples to disprove them. Some of these statements are the converse of famous theorems, others are created by omitting or changing conditions of the theorems. Some are incorrect definitions and some are seemingly correct statements. Many of the statements are related to common students' misconceptions. In this book the following major topics from a typical single-variable Calculus course are explored: Functions, Limits, Continuity, Differential Calculus and Integral Calculus.

The book can be useful as a teaching resource for:

• upper secondary school teachers and university lecturers
• upper secondary school and university students, and
• upper secondary school teachers for their professional development in both mathematics and mathematics education.

ISBN 0-476-01215-5

This book links to a larger project that Associate Professor Klymchuk is also involved in analysing the transition from secondary to tertiary education in mathematics. This larger project is supported by the Teaching and Learning Research Initiative Fund and further information aboon this can be found at www.tlri.org.nz/post-school-sector

About the Author

Dr Sergiy Klymchuk is an Associate Professor of the School of Computing and Mathematical Sciences at the Auckland University of Technology, New Zealand. He has 29 years experience teaching university mathematics in different countries. His PhD (1988) was in differential equations and recent research interests are in mathematics education and epidemic modeling. He has more than 140 publications including several books on popular mathematics and science that have been, or are being, published in 11 countries.