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Paradoxes and Sophisms in Calculus

Paradoxes and Sophisms in Calculus

By Sergiy Klymchuk and Susan Staples

Print ISBN: 978-0-88385-781-6
Electronic ISBN: 978-1-61444-110-6
108 pp., Paperbound, 2013
List Price: $41.00
Member Price: $30.75
Series: Classroom Resource Materials

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Paradoxes and Sophisms in Calculus offers a delightful supplementary resource to enhance the study of single variable calculus. By the word paradox the authors mean a surprising, unexpected, counter-intuitive statement that looks invalid, but in fact is true. The word sophism describes intentionally invalid reasoning that looks formally correct, but in fact contains a subtle mistake or flaw. In other words, a sophism is a false proof of an incorrect statement. A collection of over 50 paradoxes and sophisms showcases the subtleties of this subject and leads students to contemplate the underlying concepts. A number of the examples treat historically significant issues that arose in the development of calculus, while others more naturally challenge readers to understand common misconceptions. Sophisms and paradoxes from the areas of functions, limits, derivatives, integrals, sequences, and series are explored. The book could be useful for high school teachers and university faculty as a teaching resource; high school and college students as a learning resource; and a professional development resource for calculus instructors.

Table of Contents

I. Paradoxes
II. Sophisms
III. Solutions to Paradoxes
IV. Solutions to Sophisms
About the Authors

About the Authors

Sergiy Klymchuk is an Associate Professor of the School of Computing and Mathematical Sciences at the Auckland University of Technology, New Zealand. He has 32 years broad experience teaching university mathematics in different countries. His PhD (1988) was in differential equations. At present his main research interests are in mathematics education. He is a Fellow of the Institute of Mathematics and its Applications (IMA) based in the UK and an Associate Editor of the international journal Teaching Mathematics and its Applications. He is a member of the Royal Society of New Zealand (RSNZ). He is also a member of three affiliated groups of the International Commission on Mathematical Instruction (ICMI) of the International Mathematical Union (IMU): the International Community of Teachers of Mathematical Modeling and Applications (ICTMA), the International Group for the Psychology of Mathematics Education (PME) and the International Commission for the Study and Improvement of Mathematics Education (CIEAEM). He has more than 180 publications including several books on popular mathematics and science that have been, or are being, published in 11 countries. His book Counterexamples in Calculus, also published by the MAA, received an Outstanding Academic Title award from Choice Magazine in 2010.

Susan Staples, Associate Professor in Mathematics at Texas Christian University, traces her connections to the MAA back to her high school days in Canton, Massachusetts, where she was an active participant in mathematics competitions. She subsequently received a B.S. in Mathematics from Case Western Reserve University and a Ph. D. (1988) in mathematics from The University of Michigan in the area of complex analysis. Susan served ten years as graduate director of the TCU MAT program and continues to enjoy working with local teachers. She is the proud recipient of teaching awards from The University of Michigan, The University of Texas, and TCU. Her professional memberships include the MAA, AMS, AWM, NCTM, and the AAUW. Her graduate work was supported by an AAUW fellowship. Currently Susan holds positions on two editorial boards for the MAA — The American Mathematical Monthly and the Classroom Resources Material book series; she previously served as assistant editor for the problem section of the Mathematics Magazine. Susan has also directed the Actuarial Program at TCU since its inception. She is pleased to have been selected or the national Committee on the Undergraduate Program in Mathematics (CUPM) Actuarial Sciences group.