MAA Review of More Fallacies, Flaws and Flimflam

Fernando Q. Gouvêa reviews More Fallacies, Flaws and Flimflam by Edward J. Barbeau as part of MAA Reviews.


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.

For example, here’s a method (section 1.4) for adding two fractions with the same single-digit denominator and single-digit numerators, such as 5/8 and 1/8. First you juxtapose the two denominators to get 88. Then you juxtapose the numerators, getting 51, but of course addition is commutative, so you should also juxtapose them in the other order, getting 15. So

58+18=51+1588=6688=68. 

Neat, and perhaps an interesting one to try on students. A more advanced example is found in section 4.6, where L’Hospital’s rule is applied to

limx→∞x−sinx2x+sinx2 

 to prove that −1=1. In section 6.9, a long computation shows that

∫10x4+1x6+1dx=0.""

Read the full review here. 

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More Fallacies, Flaws & Flimflam

by Edward J. Barbeau
Spectrum Series

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. 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.

More Fallacies, Flaws and Flimflam

More Fallacies, Flaws and Flimflam by Edward J. Barbeau is now available in the MAA eBooks Store. A paperback version will be available in the MAA Store soon.

Here is another collection of Fallacies, Flaws and Flimflam, mostly drawn from the column of this name in The College Mathematics Journal between 2000 and 2008. As in the first volume, there is a variety of items ranging from howlers (outlandish procedures that nonetheless lead to a correct answer) to errors that are deep or subtle often made by strong students. While some are provided for entertainment, others offer a challenge to the reader to determine exactly where things go wrong.

There are many proposals to improve the quality of mathematics education, but they seldom address the need for students to pay careful attention to what they do and to check their work. It is through an engagement with meaning that students can avoid the pitfalls that come too naturally. Accordingly, this volume should be useful to teachers at all levels by giving them examples of flawed work they can use in the classroom. Encouraging students to find where someone else went wrong may help them avoid similar errors in the future.

The items are sorted according to subject matter. Elementary teachers will not find much of use beyond Chapter 1, while middle and secondary teachers will find items in Chapters 1, 2, 3, 7, 8 that they might use. College teachers should find material in every part of the book.

The mathematical topics covered include arithmetic, algebra, trigonometry, geometry, combinatorics, probability, and calculus.