Friday, December 20, 2013

MAA Books Beat: Calculus Online and Geometric Excursions

Written by Steve Kennedy, MAA Books Beat is a new column that will appear in MAA Focus. The first article, Calculus Online and Geometric Excursions, appears in the December 2013/January 2014 issue. 

Calculus Online and Geometric Excursions

by Steve Kennedy

My department used the same calculus text for well over a decade. We’re a traditional place, and we used a traditional text. It is a splendid book–terrific exposition with great problems. A year or so ago, yet another edition came out, and we, collectively, gulped when we saw the price tag. More than $200. We decided to stop fiddling around and look seriously at alternatives. After a long and careful comparison period during which we read and discussed a big pile of calculus books, we chose an alternate text. This text features good exposition and good problems and, more to my present point, a price tag $100 lower than the original book.

The publisher’s representative of our original text learned of our decision and the reason for it, and offered various discount prices for various lengths of time. After considering the options, we moved on to the new text. This bargaining session left me feeling annoyed and manipulated. I’m a teacher. I’m not interested in playing negotiating games. I want to help my students learn and grow, and I would like publishers to help me by providing good books at fair prices–not marketing manipulations and games.

Now I work for a publisher. A publisher that, I believe, shares my view that our role is to provide quality books, especially textbooks, at reasonable prices. This publisher also puts out the newsmagazine you are reading. My new job (just part time; I’m still a math teacher) is to acquire those quality books from you, the MAA members who are writing them.

Your probably don’t think of the MAA as a publisher, and that’s natural. We are, primarily, a collective of professionals working together to advance and promote mathematics. But producing books of enduring value is a part of that collective effort. We’re a funny kind of publisher because the people writing the books and the people buying the books are, essentially, the same: MAA members.

One part of my new job is to market those books. I’m not really sure how best to do that, although I’m dead-set against backroom bargaining and rapid edition switching designed to kill the used-book market. I prefer to calmly point out to you the virtues of some of our books. I intend to use this space on a regular basis to highlight some of the terrific books we are publishing. I intend also to, frequently and loudly, point out the following: We publish really good books at very reasonable prices. You ought to buy them.

Let me tell you about two of them.

Calculus: Modeling and Application

The first feature of Calculus: Modeling and Application, an ebook, that caught my attention was the price. For $35 your students get one year’s access to two semesters’ worth of single-variable calculus. Yes, thirty-five dollars, two semesters of calculus. I assume I’ve got your attention too.

This ebook grew out of the Calc Reform movement. Lang Moore and David Smith have reimagined the first two semesters of calculus in a way that will thrill our colleagues in the natural sciences. Calculus arises naturally out of efforts to model, to understand, the real world. Functions, for examples, are introduced by way of scatter plots of data. Differential equations (simple ones) appear very early, and physical activities include experiments and data collection to fit models. All the pedagogy is activity driven, and the activities are rich with data collection, physical experiments, and model construction and analysis. My guess is that this would be a very fun book to teach out of (and to learn out of); the activities would make for a lively classroom.

The electronic delivery allows for animated graphics to be embedded in the text. (Note that you must have a live Internet connection to read the text.) Included in the laptop version are links to Mathematica, Maple, and Mathcad worksheets. The tablet version (included in the price) delivers a terrifically smooth experience. Check out the free access to chapters 2, 5, and 8 at

Moore and Smith will be explaining the approach and demonstrating the features at a session in Baltimore during the Joint Mathematics Meetings (Friday, January 17, 3:30 p.m. in room 340 of the Baltimore Convention Center). Participants should bring a laptop or tablet for a fill hands-on exploration. All attendees will be offered 60 days’ free access to examine the text.

New Horizons in Geometry

Genius sometimes consists of seeing the simple core within a confusing, complex clutter. Every time I see John Conway talk, for example, I’m amazed anew at his ability to ask a very simple question that penetrates and illuminates the essence of a deep and complicated situation. It’s electrifying and dizzying and charming.

I have the same reaction to New Horizons in Geometry by Tom Apostol and Mamikon Mnatsakanian. It is unquestionably a work of genius. It is 500 pages of insight after insight, thrill after thrill. I will tell you two of my favorites, both about the cycloid, to give you a taste.

Imagine rolling an equilateral polygon along a line and tracking the path of a vertex during the (bumpy) motion. That path will be composed of circular arcs whose radii correspond to the lengths of the various diagonals that can be drawn in the polygon. Figure 1 shows the results of the pentagon. The authors call this curve a cyclogon and prove the beautiful theorem that the area underneath it is equal to the sum of the area of a polygon and twice the area of the disk that circumscribes the polygon. Lovely. But now imagine that the number of sides of your polygon increases without bound, and you get, in the limit, the classical theorem that the area under one arch of a cycloid is thrice the area of the generating circle.

Speaking of that classical theorem leads to another marvel in this book. We all know the classical theorem, but have you ever wondered how that area is generated? Apostol and Mnatsakanian, did, and the answer is a stunner. The area of the cycloidal sector (OPC in figure 2) is three times the area of that part of the circle has rolled so far! By that, I mean the portion of the circle bounded by the chord PC and the arc that has been in contact with the line segment. The triple are relation holds throughout the generation of the cycloid.

The book is full of wonders like this. It is not just the results that are marvelous; the proofs are even more incredible, consisting of beautiful dynamic-geometric or physical balancing kinds of arguments. It is as if Archimedes has come back to life to tell us how we ought to be thinking about geometry.

Read the book. It will take your breath away.

Steve Kennedy teaches mathematics at Carleton College in Northfield, Minnesota, and recently assumed the position of senior acquisitions editor for MAA Books. He is eager to talk to you about the book you are writing–email him at

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