The title of this book gives a clear and truthful description of what it presents. The seventy pages offer a list of some common mistakes made when solving mathematical problems and corresponding error-checking techniques, using material from calculus and algebra.

One of the most important conclusions that can be reached by reading *Misteaks* is that in our endeavors we need to use a grain a salt. Reading this booklet requires a heaping teaspoon of salt and good doses of patience and humor. But it is a worthwhile reading, and after all it is neither very time-consuming nor expensive reading. I would start by suggesting to skip the Introduction. Somehow, I find Cipra to be at his worst in it, using such a cynical tone that it is hard to decide if he is trying to be funny or if he just wants to be irritating. But then, if one skips the Introduction, one would miss the valuable remainder that "to be a successful problem solver (and I don't mean just of mathematical problems), you have to try to think in many different ways — use many different pairs of eyes, so to speak."

This is the major problem with *Misteaks* and the one for which the reader needs patience and humor. Good points are sometimes buried in the midst of so many jokes (not all funny and some right down cynical) that I felt I had to look for good ideas, of which there are plenty, by sifting through irrelevant "funny" things.

Let me give a few examples. The note on page 7 states "(You can safely ignore all parenthetical remarks in this book.)" So, why are they there? Should this remark be ignored since it is parenthetical? I found the note on page 8 annoying "...and you're wasting your time reading another footnote." And what about the suggestion "When possible, though, use symbols that will confuse anyone trying to grade your paper, then complain when they grade it wrong"? My students do not need any help with this. Just a moment, the suggestion is in parentheses... I get it! It is to be ignored. Uhm... See what I mean? Other paragraphs in the book seem to suggest that Cipra had a lot of fun writing the book, but some of them read like freely written "stream of mathematical consciousness" episodes.

So, why should we read this book? Because Cipra is a good writer of mathematical books and because this book has a lot of good mathematical stuff in it, including the following redeeming paragraph that opens Chapter 7:

Symmetry is a concept dear to the hearts of mathematicians. Inherent in nature, first appropriated by the classical geometers and artists, it now finds itself expressed in all kinds of mathematics, including the formulas of algebra and calculus. To a mathematician, there is nothing like the thrill of looking at a mathematical formula and seeing in it all the balance, harmony and sheer beauty that the subject is capable of.

It is important to make students think about harmony and beauty in mathematics and it is important for mathematicians to shamelessly advertise it.

Moreover this is a book that achieves its goal, which is to help mathematics students judge the soundness of the answer obtained for a problem and look for mistakes in procedures. Cipra lists a variety of instruments to use: common sense (however we define it), estimates, geometry, graphs, counterexamples, comparison of complicated problems to similar easier ones, analysis of dimensions, and symmetries of formulas. This book can be put to good use in several mathematics courses, therefore I am glad that the subtitle that appeared in the second edition "A Calculus Supplement" has been removed. The material presented in Chapter 5 can be used in a more advanced course when introducing the use of counterexamples. Some of the problems at the end of the chapters could be used in several occasions to get students to "grade" someone else's work and to teach them the value of careful proofreading. The suggestions given in Chapter 6, entitled Dimensions, would be interesting and familiar for engineering and physics students in any of their courses.

There is a suggestion on page 35 that I think is valuable and can be used in many occasions. Cipra is discussing the use of the second derivative test and writes

Well, if you're like me, you know that a positive second derivative means one thing and a negative derivative means the other — *but you can't ever remember which is which*... think about *x*^{2}, because *x*^{2} is a *far simpler* problem.

Then he reminds the reader that the function has a minimum at x = 0, where its second derivative is positive. The construction of a repertoire of simple mathematical objects to use as quick references is a powerful tool most students are not familiar with and do not use.

Some of the suggestions given might be a little too advanced for calculus students, but they can stimulate the instructor's imagination to find more fitting material. For example, not too many of my calculus students can visualize the graph of the function *xe*^{-x} without working on it in extensively , which would defeat the purpose of the suggestion on page 6. Similarly, few of them would use Rolle's Theorem without prompting (page 19), or appreciate the fine points of the discussion on pages 15 and 16 (they would never have doubted the size of the answer they got because they are the calculator-toting generation).

There are some statements of questionable truth, such as "A definite integral measures the area beneath a curve." (page 1) or "trigonometric functions do not spontaneously generate, nor do denominators of fractions" (page 55). And then there is the "fudging" issue. Cipra makes it clear that fudging does not always make a wrong answer correct, but he seems to suggest that when one is desperate, one could try to fudge...

I will conclude with a list of a few sentences I found remarkable (for better and for worse):

As to the stuff inside the arctan, there is a simple principle: *you can't take the arctangent of anything with dimensions*.
One other thing bears repeating: variables are so much better than numbers. You can do with variables that you could never think of doing to plain old numbers.

The essence of error-checking is to use some simple feature of the problem to predict some simple feature of the answer-and then seeing if your answer has that simple feature.

Big mistakes are easy to find; it's the little ones that'll kill you.

Read *Misteaks* and make your own list!

Antonella Cupillari ([email protected]) is associate professor of mathematics at Penn State Erie, The Behrend College. Her main research interests include history of mathematics and mathematics education. She is the author of the book The Nuts and Bolts of Proofs.