More Fallacies, Flaws, and Flimflam is the second volume of selections drawn mostly from The College Mathematics Journal column “Fallacies, Flaws, and Flimflam” from 2000 through 2008. The MAA published the first collection, Mathematical Flaws, Fallacies, and Flimflam, in 2000.
As in the first volume, More Fallacies, Flaws, and Flimflam contains items ranging from howlers (outlandish procedures that nonetheless lead to a correct answer) to deep or subtle errors often made by strong students. Although some are provided for entertainment, others challenge the reader to determine exactly where things go wrong.
Items are sorted by subject matter. Elementary teachers will find chapter 1 of most use, while middle and high schoolteachers will find chapters 1, 2, 3, 7, and 8 applicable to their levels. College instructors can delve for material in every part of the book.
There are frequent references to The College Mathematics Journal; these are denoted by CMJ.
Table of Contents
2. School Algebra
4. Limits, Sequences and Series
5. Differential Calculus
6. Integral Calculus
8. Probability and Statistics
9. Complex Analysis
10. Linear and Modern Algebra
About the Author
Excerpt: 5.8 A standard box problem (p.89)
Dale R. Buske put on a recent calculus examination the following standard problem:
Problem A crate open at the top has vertical sides, a square bottom, and a volume of 4 cubic meters. If the crate is to be constructed so as to have minimal surface area, find its dimensions.
One student started with this formula for the surface area of the crate: SA = 4x + 4y, where x was the length of one side of the base and y was the height of the crate. (After all, there are four line segments of length x on the bottom of the crate, four line segments of length y on the sides, and the four line segments on top do not count since the crate has an open top.) The student then correctly went on to use the volume constraint 4 = x2y to find y = 4/x2 and arrive at the formula SA = 4x + 16/x2. Taking the derivative and applying the condition for a maximum leads to the correct answer: the crate should have a base with dimensions 2 meters by 2 meters and a height of 1 meter.
About the Author
Ed Barbeau graduated from the University of Toronto (BA, MA) and received his doctorate from the University of Newcastle-upon-Tyne, England in 1964. He was a Postdoctoral Fellow at Yale University in 1966-67, and taught at the University of Toronto from 1967 until 2003. He is currently retired.
Barbeau was named Fellow of the Ontario Institute for Studies in Education (1989), and has received the David Hilbert Award (1991) from the World Federation of National Mathematics Competitions and the Adrien Pouliot Award (1995) from the Canadian Mathematical Society. He has published a number of books with the MAA as well as two books, Polynomials and Pell’s Equation with Springer. He has been invited to give talks frequently to groups of teachers, students and the general public, and has presented three radio broadcasts, “Proof and truth in mathematics,” in the Canadian Broadcasting Corporation series, Ideas.
Some fifteen years ago, I reviewed Mathematical Fallacies, Flaws and Flimflam, the first book collecting the best of Edward J. Barbeau’s regular column in The College Mathematics Journal. Much of what I said then applies here as well: this is an entertaining book that can also be useful in the classroom.
The typical FFF item gives an example of a mathematical argument that is wrong but tricky. Sometimes the problem is that the argument, while visibly (even extravagantly) incorrect, gives the right answer. Other times, the argument contains a subtle error, or uses a method that is correct for unexpected reasons. Continued...