Devlin's Angle

April 1997

When mathematics has to be theater

A tale with a moral

Several years ago, I promised myself I would never write another textbook. Not that I was losing interest in mathematics or in writing. Rather, it was getting to the point where there were so many good textbooks around, I could no longer see any place where I could make a significant new contribution. At least, that is how I saw -- and see -- the situation for "specialist" textbooks in my area, aimed at mathematics majors. When it comes to introductory textbooks such as pre-calculus and calculus, I see a different state of affairs.

These days, calculus textbooks come in two kinds: hard covered, called "traditional", and soft covered, called "reform". The former favor cover designs featuring intellectually inspiring items such as violins; the latter prefer a more sporty image, such as a hot air balloon. Of course, I jest. The two kinds of textbook really do differ, and I know from experience that writing a good textbook is no easy matter. Within each of the two calculus camps, however, the traditional and the reform, the contents of one textbook differ little from any other. The last time I was involved in making a departmental textbook adoption decision, the final choice between maybe half a dozen texts was made almost entirely on the basis of design: which textbook did we think the students would find the most attractive and the least intimidating?

So when, eighteen months ago, I was approached by a publisher and asked to write a calculus textbook, I had no hesitation in saying "no." Even though this was not to be a paper textbook but an electronic text on CD ROM. The publisher persisted. Interactive texts are not at all like traditional texts, she insisted. An interactive text is not -- or rather should not be -- just a book transferred onto a disk and read on a computer screen.

"Why not approach one of the established calculus textbook writers," I retorted, even going so far as to name several whose work I admired. "No, I think it is important to have someone who has not written a calculus textbook before," the publisher replied. "We want an author whose first attempt to write a calculus text is to write it as an interactive text. Done properly, an interactive text should be very different from a book. The entire presentation is different. You have to re-think everything from the ground up."

I was still not convinced, but I was certainly intrigued. As readers of my recent book Goodbye, Descartes will know, my research interests for the past several years have been in language, thought, and human cognition. Trying to figure out how to use multimedia to help students learn calculus seemed like an interesting sideline to that research project. During my career, I have worked on projects to explain mathematics in the newspapers, on radio, and on television, and have always enjoyed the challenge of trying to use a new medium to get the message across. Getting a newspaper article past a hard-nosed news or features editor is very different from getting a piece into, say, the American Mathematical Monthly. And radio and television, whether scripted or "live and spontaneous," are very different environments from the classroom or the mathematics department common room. What new challenges would arise in trying to use multimedia to explain calculus?

On the other hand, I knew of a number of projects nationwide to develop interactive texts on a variety of topics, including calculus. Why didn't the publisher just approach one of them and offer to publish their product commercially, I asked. After all, for those projects, most of the initial development work will have been paid for by grants or institutional support. In contrast, I would have to start from scratch. This time, the publisher's answer was, for me, almost the clincher. Her intention was, she said, to design and develop the product as a commercial product, with all the pressures that would entail. It would be sold directly to students (not sold by class adoptions), and hence would have to appeal to students on its own merits. It would have to sell for under $30. It would have to run on any reasonably contemporary PC or Mac. It was not intended as a "first text;" it would be called an Electronic Companion to Calculus, designed to complement any college-level first-year course, regardless of the textbook used or the method of instruction. It would have to be designed and produced on a tight budget built into the company's business plan. These were just the kinds of issues I had faced writing for the press, radio, and television. Creating a good, honest product under those kinds of constraints presents a real challenge I had found I enjoyed meeting.

As I said, I was almost hooked. I had just one final question. The project would clearly be a joint venture. I know about calculus, and like every mathematician I have an opinion as to how it should be taught. But I knew little about developing multimedia and about using interactivity as a pedagogic device. Who would be the other members of the team? They would be crucial to any possibly success. So the publisher arranged for me to meet the others. Suffice it to say that after I had met them, I was highly flattered to have been approached to be the author. This was an impressive and highly talented team, with a wide variety of experience. We were about to enter uncharted waters, and the final product might "bomb." (Few CD ROM products, if any, have been commercial successes to date, and some major publishers are already getting out of the CD ROM business.) But working with that team would be a marvellous experience, regardless of the outcome.

So I said yes. And promptly found myself living the kind of life generally associated with twenty-five year old computer whizzes working for Silicon Valley startup companies. With a full-time job as dean during the day, burning the midnight oil became my way of life for several months, as we deconstructed calculus and reassembled it from the ground up so that everything -- both mathematical content and all the navigational apparatus of multimedia -- could fit onto one screen. Whatever the topic, I had to make do with one small text area and one slightly larger image area. Paging forward was, my interface expert kept telling me, to be avoided wherever possible. Don't use words, use diagrams, animations, and interactivity, she kept saying. Don't present the material in a linear, cumulative way, she insisted. The user will want to jump around and explore, using the hyperlinks, very likely not starting at "screen 1" at all.

Using interactivity was the thing I found hardest to get the hang of, a sure sign of my half century age. Like most of my generation of instructors, I tend to think in terms of content, of explaining the method, and of conveying information-as-a-commodity. Interactivity is about process. Toward the end of the project, I began to appreciate the potential of interactivity as a pedagogic device. It really is very different from anything else in the instructor's arsenal. Given a fresh start and a very large development budget, I think I could now take real advantage of what interactivity can offer. But as often in life, we had to live with what we had. We had all but spent our development budget and the business folk were pressing us to get the product out to market. For me this was an academic exercise. For everyone else, it was a business, and their livelihood depended on it!

Last week, I finished the project. For good or bad, the thing is ready to go on sale. Am I happy with it? Well, it's not the product I would now like to produce, given what I know now. With lots more time and a much bigger development budget, I could really go to town! On the other hand, I have to say I do like it. I think we really have managed to take advantage of the medium to produce something new and useful. It does not -- and was never intended to -- replace anything else on the market. It was conceived from the beginning as a companion, designed to complement everything else that is available.

I believe it definitely has a "personality," a very important feature for a stand-alone product of this nature, and a feature we struggled hard to develop. Though my name appears as the author, the final product represents the joint creative work of Belen, Bruce, Carol, David, Marian, Travis, Trish, Volker, and many others.

The point of this story? Not to advertise the product. The company has a marketing plan, and they don't need my help in that arena. (Besides, the product is designed to sell directly to students, not to the faculty who I assume are the main readers of Devlin's Angle.) Rather, my point is to say that if anyone is eager to develop multimedia educational products, you should get involved with professionals experienced in the area of multimedia before you put pen to paper (or finger to keyboard). One thing I learned early on, and kept being reminded of again and again, was how complex are the technical and cognitive issues that arise in multimedia. A multimedia product is not just a book and a few animations put onto a CD ROM. It's a form of solitary experiental theater, and it has to be developed in that way.

I know we did not get everything right. I have ideas how we could improve lots of things; and I am certain there are flaws none of us are yet aware of. But on my own, with just my knowledge and experience as a mathematician and a writer, I could not have produced anything remotely like our final product. Writers of textbooks often complain about being "pressured" by editors to present (or delete) material against their will, or to change the presentation. With multimedia, those kinds of pressures are much more intense.

The mathematician who enters the world of commercial multimedia should be prepared to be just one cog in a very large and complex machine. To a great extent, the mathematical integrity of the final product rests largely on the author's shoulders, and once the thing is out there, there will be no shortage of critics. So if you enter this new world, you will find yourself struggling to find new ways to get an old and familiar message across. In the case of an educational product, the medium is definitely not the message, of course. If there is something that you, as author, think has to be covered, it will be up to you to find a way to get it in that passes the muster of the others on the team, whose concerns will be with implementation, attractiveness, look-and-feel, ease-of-use, overall "theatrical" structure, cost, and marketability. When push comes to shove, it's amazing what can be presented on a single interactive screen!

- Keith Devlin

Devlin's Angle is updated at the beginning of each month.

Keith Devlin ( is the editor of FOCUS, the news magazine of the MAA. The CD ROM package Electronic Companion to Calculus will shortly be published by Cogito Learning Media, Inc.