Book Review…Using Counter Examples in Calculus, John Mason and Sergiy Klymchuk. Imperial College Press, 2009, ISBN-13 978-1-84816-360-7(pbk)

In life it is so easy to have too much of a good thing. This is how I felt after a first reading of “Using Counter Examples in Calculus”, by John Mason and Sergiy Klymchuk. The heart of this book is five pages of exercises for students arranged in five groups, functions, limits, continuity, differential calculus and integral calculus. However, what is unusual is that each exercise is an incorrect mathematical statement, to which the student must find a counter example. 3.4 is typical: “A function always has a local maximum between any two local minima”. An expert might dismiss such examples as trivial, their familiarity is, after all, a hallmark of their expertise. The student might well find constructing such a counter example a significant challenge. The point of the book is to provoke teachers to take such examples more seriously, to use them in their teaching and as a result to encourage students into a mathematical playfulness through which they raise conjectures for themselves. For example, I rather dismissed 3.12 with a constant function. 3.12 If a function is continuous on [a,b] then it cannot take its absolute maximum or minimum value infinitely often.

Well, the constant function isn’t the solution given in the very detailed discussion of this statement. Their example has different maximum and minimum values, not a requirement of the given statement, but perhaps the sort of supplementary question one might choose to use with a group of students in a discussion. Is such a function necessarily constant? The bulk of the book is a detailed discussion of each statement, with a variety of counter examples and further constraints or conjectures to consider.

All this moves away from giving the answer the teacher wants, or seeing the answer as unique, to building confidence with these basic concepts. All the statements are carefully chosen, based on educational research and experience of teaching calculus/analysis. I recognized among them many of the misconceptions students cling to. The book also acts as a very useful catalogue of such examples and statements, for example when planning such a course….