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Using Counter-Examples in Calculus

John Mason and Sergiy Klymchuk
Imperial College Press
Publication Date: 
Number of Pages: 
[Reviewed by
Henry Ricardo
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Both the book under review and Klymchuk’s Counterexamples in Calculus (MAA, 2010), which I bought at the last Joint Mathematics Meetings, are reincarnations of a book published in New Zealand in 2004. Almost every assessment in the review of the MAA version can be applied to the Mason-Klymchuk volume.

Although the authors state that their book can be used as a teaching resource, a learning resource, and a tool for the professional development of upper secondary school teachers, I believe that this book would be more useful to teachers than to the average student. It includes a larger bibliography of articles and presentations on education than the MAA edition. On the other hand, the MAA editors seem to have suggested the insertion of references to contemporary American books in their version. The first two chapters (missing in the MAA book) establish the pedagogical foundation for the study of counterexamples, providing case studies by the first author, quoting a number of mathematical scientists, and citing various educational studies. According to Mason and Klymchuk, students should work with counterexamples 1. For deeper conceptual understanding; 2. To reduce or eliminate misconceptions; 3. To advance mathematical thinking; 4. To enhance generic critical thinking skills; 5. To expand the “example space”; 6. To make learning more active and creative. (The authors suggest speaking and writing of “example spaces” as an alternative to discussing “pathological examples.”)

There are minuscule differences between the MAA book and the book under review in the actual statements for which counterexamples are sought and provided. In addition, a central criticism of the MAA book has been weakened: The Mason-Klymchuk exposition asks more pedagogical questions, such as “What is it that makes the example work as a counter-example?”, “What additional assumptions about f are needed to make the conjecture valid,” and urges the reader to “Generalise.”

The British edition seems suitable for teachers of calculus, whereas the American edition is closer to supplementary reading material for students. Either book should have a place on a calculus instructor’s book shelf.

Henry Ricardo ( has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.


  • Working with Counter-Examples
  • The Pathological Debate
  • Bones to Chew: Collection of False Statements
  • Suggested Solutions and New Challenges