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Click It! Drag It!

Alexander Bogomolny

In the inaugural issue of JOMA, David A. Smith, Editor of JOMA, enumerated the many reasons why the new undertaking has a good chance to succeed where its predecessor, Communications in Visual Mathematics (CVM), did not. I concur with the reasons and sincerely hope that anticipation will come true. I would have been more content had an intent for aggressive promotion (the expression I borrow from the original NSF proposal) also been mentioned. Let's hope that promotion has not been simply overlooked but rather has been put off till a later stage, say till the library reached some level of completeness.

About the author: Alex Bogomolny writes the Cut the Knot column for MAA Online and is the developer of the popular Interactive Mathematics Miscellany and Puzzles site.

When may it be in any sense complete? In my view, completeness requires two ingredients: quality and comprehensiveness. As with a heap of grain that preserves its essence of being a heap even when small quantities of grain are being added or removed from it, so with comprehensiveness: a library may be called comprehensive when it covers most of the topics in a specific course, give or take a few. Put another way, a library is comprehensive when using it in a course becomes (at least potentially) a matter of routine, rather than exception. You'll probably know it when the time comes.

I can't dispense as easily with the matter of quality. First, the notion of quality applies to a single item in a library as much as to the collection as a whole. Second, quality is very much in the eye of the beholder. Talking of quality is almost like discussing tastes. What's an eye-catcher for one is a noisy distraction for another. G. C. Lichtenberg characterized good taste as either that which agrees with my taste or that which subjects itself to the rule of reason. What follows is my two-cents-worth on the subject of quality. Needless to say, one's judgment is tinted by one's tastes. To bestow an appearance of objective reasoning on the ruminations below, I'll use as scaffolding Robert Devaney's review of the Interactive Differential Equations CD, IDE for short [American Mathematical Monthly 105 (1998), 687- 689].

The review opens with an engaging paragraph: "Every once in a while a software package surfaces that absolutely revolutionizes how a particular course or a series of courses is taught. Such is the case with the Geometer's Sketchpad and other interactive geometry packages ... I consider the Interactive Differential Equations (IDE) software package to be no less revolutionary ... ".

The IDE is not an equation solver but rather a collection of nearly one hundred different "tools" for teaching and learning differential equations. Each tool has one and only one purpose: to illustrate a single idea involving differential equations. Understanding the relationships among multiple representations and visualization of solutions of differential equations are the goals of each of the tools in the package. On the whole, the review is very enthusiastic about the IDE. As an instructor, Devaney does not seem to mind that some differential equations have been omitted. He just suggests employing other ODE packages if necessary. The only complaint we find in the review refers to several tools that involve five different representations of an ODE, all evolving at the same time -- too much of a good thing. Even when the instructor wants students to watch only two of the graphs, some look at other animations. It would have been nice to be able to turn individual displays on and off as needed.

On the other hand, a few tools from the IDE present topics usually not included into an ODE course as too difficult. With the tools, students really "get it." The tools can be used in demos, but also in the lab for student investigation.

One point has not been mentioned in Devaney's review: the consistency of user interface. In all likelihood, the reviewer took it for granted. After all, the tools not only came as part of a single package, but they have all been designed by the same team of four mathematicians and an artist.

Two points I believe are very easy to agree on. First, it's reasonable to think of the IDE as a topical digital library on CD, something like that initial portion of the MathDL which is devoted to Calculus mathlets. Second, JOMA staff would probably love nothing better than getting a review of the MathDL in the spirit of Robert Devaney's review of the IDE. That review has an immediate bearing on the questions of quality of individual mathlets and of comprehensiveness of the collection. I shall offer my intuition regarding quality of the collection as a whole.

A mathlet is by definition a small interactive web-based tool for use in teaching (and, of course, learning) mathematics. The idea that a mathlet might revolutionize how a particular course is taught is incongruous. Adding a mathlet to an existing collection should not make any difference either. In time, however, the collection will acquire a critical mass with the potential to affect a whole course of instruction. Realization of the potential will depend on the quality of the collection. In this respect, a library of mathlets gleaned from all over the Web is at a great disadvantage compared to a predesigned package of software tools: Library mathlets are liable to sport different user interfaces and multiple technologies.

I am reminded of an essay Of Custom by Michel de Montaigne. "He seems to me to have had a right and true understanding of the power of custom, who first invented the story of a country maid who, having accustomed herself to play with and carry a young calf in her arms, and daily continuing to do so as it grew up, obtained this by custom, that, when grown to be a great ox, she was still able to bear it."

The clever students and instructors have a high degree of adaptability and can easily accustom themselves to every other way of acquiring and processing information. But, once a custom has been developed of doing that in a specific way, they become stiff with resistance to do things differently.

To sum up, there is a valid concern about sharing design and user interface among items in MathDL. A public discussion addressing this concern will be genuinely welcome.

Let's now turn to individual mathlets. More specifically, I wish to talk about so-dear-to-my-heart Java applets. As with other programming languages, in Java there is usually more than one way to accomplish the same thing. And I am sure that some ways will be found more convenient than others by a greater part of the user community. The question is which? Consider such a simple task as inputting an integer. In the first mathlet collection there are two applets that demonstrate convergence of Riemann sums. The number of subintervals in one is controlled by a scroll bar -- in the other it must be typed in. Are there any preferences?

No less important than implementation is a question of pedagogy: what does it take to illustrate the meaning of the definition of Riemann integral? Integrals of how many functions should be approximated? One applet offers a selection of nine functions, the other allows the user to type in a great variety of algebraic and transcendental functions. Joining the fray, I wrote another applet. Based on cubic spline interpolation, there's a great deal of modifiability but, from the user perspective, none of the functions, except perhaps for the very first one, which is a parabola, is defined explicitly. Is it important that they be?

On the positive side, this applet, as does its predecessor, displays the integral not as a single numeric value but as a function of the upper limit. In addition, it displays the linearly interpolated Riemann sums, which provides some complementary information about their convergence for functions of different shapes. It's perhaps edifying to observe how multiple errors cancel out in the final number.

Be that as it may, there are so many ways to present an idea, and then so many ways to implement it as a mathlet, that the developer inevitably faces hard choices. As Montaigne wrote in another essay (How Our Mind Is Hindered By Itself), "It is an amusing fancy to imagine a mind exactly balanced between two similar desires. For it is indubitable that it will never make a choice, because comparison and selection imply inequality of value; and if we were placed between the bottle and the ham, with an equal appetite for drink and food, there would be, doubtless, no help save to die of thirst and hunger."

The user will eventually face many choices as well. And this brings us to a third issue related to the user interface, that of JOMA and MathDL themselves. Faced with many choices, user may freak out and make no choice at all. No single mathlet or even the idea it illustrates may be sufficiently important to justify a prolonged deliberation. The interface should take the initiative of guiding and assisting the user. How best to do this is also open to discussion. Having a look around and learning how things are done on the Web could be a necessary first step. For example, the manner in which books are recommended at is definitely worth paying attention to.

The bottom line is that JOMA is not a usual publication. Being online at this time engenders possibilities that could not be contemplated in the world of publishing just a year or two ago. Proactivity is a necessary component of a successful enterprise. Active promotion is only a part of it. Nowadays, intelligence and proactivity are also expected of user interfaces. Just think of that.

Let me finish with another quote from Montaigne (A Usage of the Island of Cea): "If to philosophize be to doubt, as they say, then with stronger reason to treat matters ignorantly and fancifully, as I do, must be to doubt. For it belongs to scholars to examine and discuss, and to the rightful judge to determine." Which you undoubtedly will.

Published July, 2001
© 2001 by Alexander Bogomolny

Alexander Bogomolny, "Click It! Drag It!," Convergence (November 2004)