Review of More Fallacies, Flaws and Flimflam

Tuesday June 10, 2014

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Cover of More Fallacies, Flaws and Flimflam
Cover of More Fallacies, Flaws and Flimflam

My review of More Fallacies, Flaws and Flimflam by Edward J. Barbeau:

Fallacies, Flaws, & Flimflam is a column regularly appearing in the College Mathematics Journal (CMJ) that highlights remarkable errors made by high school and college-level math students. These errors may be errors in proofs that leads to incorrect results or unusual assumptions that lead students in the wrong direction. These errors are often humorous, usually subtle, and frequently touch on complex reasoning that may be eluding other students or practicing mathematicians, alike. More Fallacies, Flaws & Flimflam is the second collection of these columns focusing on material published from 2000 to 2008 in CMJ, following up on Mathematical Fallacies, Flaws and Flimflam published in 1999.

The book is well-organized with eleven chapters broken down by high-level subject area, such as combinatorics, geometry, and statistics. The chapters proceed in a sequence roughly mirroring the course of mathematics instruction and each chapter contains between six and thirty individual examples of flimflam. Credit is given to the source when known, but there is little setup and context beyond the immediately necessary to understand the problem. This detracts from some problems described, such as flawed questions from math olympiads, where understanding the resolution would provide some humanity to the situation described.

In addition, beyond the high-level chapter groupings, no connection is made between examples. This makes it easier to read the book by filling picking it up and turning to a random page to fill a five-minute void in time. However, it makes it difficult to sit down and read through the text. Given the amount of time necessary to work through some examples, this may be reasonable.

The flawed examples usually include a statement that seems obviously correct but leads the student to an incorrect results. More interesting are the cases where the example includes an obviously incorrect statement, but still leads to a correct result. The book does not provide many explanations and finding finding why the incorrect reasoning worked in the example is part of the good challenge the book presents.

Many of the included flawed examples, will seem familiar since the underlying error in mathematical reasoning is not uncommon. However, the book is not useful, in the general sense, for application in the classroom. Advanced students may enjoy the exercise of breaking down the flawed reasoning and correcting it. But with few exceptions, the problems contained probably should not be given to students, who are likely to be tripped up by the same errors the students described did.

More Fallacies, Flaws & Flimflam is a delight to read, slowly, for the mathematically inclined but understanding the problems requires concerted effort across the entire range of mathematical subfields. The book would not be appropriate reading material for those with a casual interest in mathematics, but might present a series of interesting challenges for a younger but advanced student. Keeping pen and paper on hand while reading is also advised.