Mathematics for the Liberal Arts

Jason I. Brown
Publisher:
Chapman & Hall/CRC
Publication Date:
2015
Number of Pages:
440
Format:
Hardcover
Price:
89.95
ISBN:
9781466593367
Category:
Textbook
[Reviewed by
Richard J. Wilders
, on
02/19/2015
]

This is a textbook designed for use in a general education mathematics course for students with no specific skill needs in mathematics. It is part of a new series of textbooks being produced by CRC publishing. One of the goals of the series is to provide quality texts at affordable prices. I would argue that this book fulfills that promise: as of February 9, 2015, its price on Amazon.com was $80.94 — a bargain for a hardcover textbook. There is a free ebook copy available for download with a password provided in the front of the book. I assume this would only allow one download, so that students who purchased the book used would not have access to this resource. Mathematics for the Liberal Arts is a first-rate text. It invites students into the world of mathematics with well-chosen (and well-told) vignettes which appear at the start of each chapter. The topics are well-chosen as well. They include some of the old standards for these courses as well as several new topics. The author is also a blues musician who credits his work attempting to recreate the opening chord in the Beatles’ tune “A Hard Day’s Night” with providing the inspiration for this book. You can hear him play blues guitar and describe how he views the interactions between mathematics and music at Jason I. Brown Homepage. The introduction to Chapter 7 tells the tale of how “A Hard Day’s Night” came to be. Texts for liberal arts (or quantitative reasoning) courses tend to fall into one of two broad areas: courses which attempt to show the usefulness of mathematics in a range of areas and those which attempt to show the beauty of mathematics as a creative discipline. Mathematics for Liberal Arts strikes a nice balance between these two, offering several chapters of quite practical mathematics (Health, Safety and Mathematics, Making Sense of Your World with Statistics, Money and Risk) followed by an interesting sampling of less applied topics, including chapters titled Art Imitating Math, Mathematics of Sound, and Late Night Mathematics. As you might guess from Dr. Brown’s night job, the chapter on Mathematics of Sound is very well done indeed! With the exception of chapters 4 and 5, the chapters are largely independent of one another, allowing for considerable flexibility in designing a course based on this book. I would suggest including a sampling of the first 5 chapters as well as chapters 6, 7, and 8. All of this material has appeared in other liberal arts math texts, but not (so far as I know) in quite such a nice format. For that reason, I am providing a summary of each chapter. If that’s too much information I can tell you at the outset that this is a wonderful text that should appeal to a wide range of students. Details below! One of the book’s many strengths is that each chapter is introduced with a short vignette which introduces the topic at hand. Chapter 1: Health Safety and Mathematics is introduced with a story about a commercial aircraft which runs out of fuel midflight. The introduction describes the pilots calculating the amount of fuel needed for the flight and taking off confident in their result. At the end of the chapter we learn that they had miscalculated a conversion. As a result they added only 4926 liters of fuel when they in fact needed 20,092 liters! The chapter itself is a nice treatment of the process of converting units and includes an example involving which size pizza to buy to get the best price per pound. It also has a telling example of a woman who died because her nurse miscalculated the drip rate for an intravenous medicine. They administered the medicine over 4 hours instead of 4 days! This is not a topic I would have thought of for such a course, but the two cautionary examples (and the pizza calculator) convinced me otherwise. Chapters 2 and 3 deal with statistics and the visual representation of data — standards for a book of this type. As with chapter one, however, the opening example grabs our attention. In this case it is the now infamous tale of Dr. Andrew Wakefield. Dr. Wakefield published a seriously flawed study claiming a relationship between MMR vaccines and the development of autism. Once again, we are told only the first half of the story prior to the chapter. Surely a drug which induces autism is to be avoided, right? Wrong! At the end of the chapter (with the statistical knowledge at hand to understand the issues) we learn that Dr. Wakefield used an n of only 12 and that most of “evidence” was anecdotal. We learn in addition that Dr. Wakefield had been paid roughly$800,000 to provide evidence that the MMR vaccine was harmful. If you don’t think twice about statistical studies after reading this tale, I have a water-front property in Death Valley I would like to sell you.

These two chapters provide as nice an introduction to statistics (at this level) that I have seen. Chapter 2 concludes with a list of “garden paths” statistics can lead you down: sampling bias, interviewing bias, people sometimes lie, and others. This is really well done. If students remember nothing else about statistics but these cautionary notes, they will still be better able to deal with the myriad times we are bombarded with numbers in an attempt to sell something or garner our support for a candidate.

Chapter 3 is a really nice introduction to the graphical representation of data, complete with lots of very well-done graphs. Its vignette concerns the creation of the Google page rank algorithm. The chapter begins with pie graphs, bar charts, scatterplots, and the like and ends with relationship graphs (who knows who) and Euler paths — two other standard topics in liberal arts texts. These are also quite well done, including Prim’s algorithm for finding a minimum spanning tree and Dijkstra’s algorithm for finding the shortest paths from one vertex to all the others in a directed or undirected graph. There are several very nice examples. Of course, no chapter on graphs would be completed without the Kevin Bacon graph, and there he is on page 153! My only complaint — why not the Erdős graph as well?

Chapter 4, on Money and Risk, covers the theory of compound interest as well as an introduction to probability in a short 55 pages. I suspect this may be too brisk a pace for many students in a course such as this book designed to serve. The sophistication required is also significantly higher than needed for the earlier chapters. That being said, the diagrams illustrating the calculations are very well done. Students willing to put in the work should be able to work their way through the details.

Learning to Make the Best Decision starts out with a discussion of expected value — material which requires the probability material from Chapter 4. The highpoint for me was a discussion of the PowerBall. It turns out that if the prize is $250,000,000 the expected value of a$2 ticket is $1.79! Of course the announced prize is never what you actually get. I think the author missed a chance to review the previous chapter by comparing the announced prize of$250,000,000 with the value of the annuity which is offered in lieu of a single payout. This chapter also includes game theory and a nice discussion of voter paradoxes including a section titled “An Arrow to the Heart of Voting”. It’s a fun chapter with lots of good mathematics; it would form the core of the course if I were using this text.

Chapter 6, on Art Imitating Math, opens with a tale of a young known as “Mauk” or just “MC” who loved art at school but wasn’t so good at math. He visited the Alhambra and was impressed with what he saw but thought the abstract nature of the designs he saw made them less exciting than they could be. A few years later his brother gave him a paper by George Pólya and … M.C. Escher was on his way! We are then given a wonderful tour of the world of art and mathematics including the golden section, tessellations, the types plane symmetries and fractal images. The exposition is inviting and clear, the illustrations are first-rate.

Chapter 7, Mathematics of Sound, opens with the story of the Beatles’ iconic “It’s Been a Hard Day’s Night” and segues into a discussion of sound as sine waves. Included are:

• a discussion of the how the overtone series explains why we can easily distinguish a trombone from a trumpet,
• a discussion of tuning and the well-tempered tuning system invented in Bach’s time,
• the mathematics of composing — including a quote from Bruce Springsteen explaining that Beatles’ tunes are still beautiful because “…the mathematics in them is elegant.” (Who knew The Boss knew math?),
• And, you knew it was coming! A section entitled “Why Are the Blues so Damn Good?”

This chapter is a real joy to read. It is a really first-rate discussion of topics which should be of interest to most students. Highly Recommended!

Finally, we reach the end — in this case a final chapter which (very) briefly considers mathematical paradoxes, Mel Brooks, and well, read it for yourself — it is lots of fun.

To sum up: I found this to be a very fine liberal arts mathematics text. It is well worth your consideration.

Richard J. Wilders is Professor of Mathematics at North Central College. He is interested in the history and mathematics and of science and teaches courses in both areas.

Health, Safety, and Mathematics
Found in Translation
The Essentials of Conversion

Making Sense of Your World with Statistics
Summarizing Data with a Few Good Numbers
Estimating Unknowns
Leading You Down the Garden Path with Statistics

Visualizing with Mathematics
Seeing Data
A Graph Is Worth a Thousand Words

Money and Risk
Money — Now or Later
Risk Taking and Probability

Deciding to Make the Best Decisions
Making the Right Choices for You
Game Theory — Coming Out on Top
Making Joint Decisions

Art Imitating Math
The Math that Makes the Art
Believing What You See (or Not)

The Mathematics of Sound (and the Sound of Mathematics)
The Mathematics of Listening
The Mathematics of Composing
Solving Musical Mysteries with MSI (Math Scene Investigations)

Late Night Mathematics — Humor and Philosophy
Laughing with Mathematics
The Limits of Mathematics

Bibliography

Index