Journal of Online Mathematics and its Applications

The JOMA Mathlet Project

by Tom Roby

Welcome to the Inaugural Issue!

Welcome to the inaugural issue of JOMA Mathlets! This regular section of JOMA will present some of the best small interactive web-based tools ("mathlets") for use in teaching mathematics. We expect that many readers will be able to make immediate use of these mathlets, some in classroom demonstrations, some for student exploration.

In this first issue we focus on calculus mathlets. These are not all the good calculus mathlets around, or even all the good ones we found. But they are a sampling of the best that are currently available. More information on the search and review process is available in the next section.

Interactivity is key not only to the value of mathlets, but also to JOMA itself. Please join in the discussion. If you find an innovative way of using a mathlet, or an ingenious way of breaking it, add your comment directly to the mathlet's page. This will be of great service to the author, to others trying to use the same mathlet, and to potential authors who may want to improve on what has been done before.

Indeed, in the world of mathlets the biggest issues currently are high redundacy (duplication of effort) and a need to promote quality. There are many different versions of such a basic mathlet as one that demonstrates Riemann sums. And (as Jerry Porter is fond of pointing out) these same sorts of tools have been around in one form or another as long as computers have been able to draw pictures. The big difference now is our freedom to easily share things globally. By highlighting some of the best mathlets currently available, we hope to provide both an incentive for creating high-quality work, and standards for authors to meet and improve upon.

Seeking and Refereeing Mathlets

The JOMA Mathlets Project (nee "Applets Project" before we adopted the term "mathlet") was funded as part of the NSF Digital Library 2 initiative. As such, it is a child born before its parent, the MathDL Project, which was funded the following year under the NSF National Science Digital Library Project. The idea of JOMA as an online journal was central to the Mathlets Project from its inception, but it had no base of support. Fortunately, MAA interest and NSF funding have allowed MAA to create the journal as part of the larger MathDL context. The Mathlets Project funding has in turn given JOMA a jump start in mathlet collection, testing, and reviewing, along with the construction of some of the necessary infrastructure.

A team of five students worked for ten weeks in the summer of 2000, mining known collections and using search engines, to harvest 622 potential mathlets. They were asked to do a rough cut, keeping only what they thought would be useful. Though they might have eliminated some applets of value, this can always be corrected later through the normal submissions process. In practice, they kept many things that the faculty reviewers, who came later, easily eliminated from the publishable pool.

The students tested whether the mathlets work on both Macintosh and Windows platforms, running Internet Explorer and Netscape, and categorized them according to "generic tables of contents". The latter were created at Swarthmore by Gene Klotz and Steve Maurer to make it easy to browse mathlets by topic. (Although the AMS has classification schemes that are useful for research, no schemes were handy for dividing up the undergraduate mathematics curriculum.) About half of of the sites the students collected fell within the calculus table of contents; most of those remaining were precalculus, with some advanced courses also represented.

[Editor's Note (2004): The generic table of contents has been replaced for classifying Mathlets by the Subject Taxonomy -- see the link on the JOMA Home Page.]

In late August a retreat was held at the Math Forum in Swarthmore, where eight faculty from a variety of institutions and backgrounds gathered to review the calculus mathlets. Although we never achieved consensus on all the definitions and reviewing criteria, we eventually converged in the neighborhood of a local maximum. It was much easier to do this while actively engaged in deciding what mathlets would be good to publish. The next section has our current review criteria.

Another component of the Mathematical Sciences Digital Library, which includes JOMA, is the less-stringently-refereed Library of Online Mathematics and its Applications (LOMA). LOMA will provide a repository for mathlets that seem useful but may not be up to the standards of JOMA. The same general criteria are used but not applied as stringently. As just one example, a well-designed mathlet that only runs in one browser or under one operating system would be accepted to LOMA if the author did not want to make changes.

[Editor's Note (2004): LOMA was later renamed Digital Classroom Resources (DCR).  To access DCR, return to the MathDL Home Page.]

Many of the sites harvested by the students were clearly unsuitable for publication as mathlets, having been designed with some other purpose in mind. No slight is intended against any of these authors, most of whom still have no idea that their work was ever scrutinized for fitness in a journal they never heard of. For those mathlets that we did find suitable for publication, we attempted to contact the author(s) directly and ask whether they would care to submit their work to JOMA. Most authors responded positively, allowing us to publish their work.

In section 4 you will find comments made by some reviewers describing mathlets that proved unsuitable for JOMA.

In the future we expect -- like most journals -- to get the bulk of our submissions from authors before we review them. But we will also happily take recommendations from the math community of sites that are worth pre-reviewing. In the short term, we plan two more web-harvesting expeditions, for issues of JOMA focusing on precalculus and statistics.

JOMA Mathlets Review Criteria

In our summer 2000 reviewing effort, we started from Gene Klotz's definition of a mathlet:

A mathlet is a small, interactive, platform-independent tool for teaching math -- the equivalent of a good example that you want to haul out, give (or show) to your students, and let them go explore. Some will be more general purpose and of broader use, but the basic idea is that they should be simple to explain and to use.

Although technological specifications are bound to change with the times, at the moment we expect submissions to run on both Windows-based PC's using Netscape or Internet Explorer, and on Macintoshes using Internet Explorer. (The current version of Netscape for Macintoshes uses a very old version of Java, so most applets will run poorly if at all.) Plug-ins may be used, provided that they may be downloaded without cost.

JOMA REVIEW CRITERIA

Essential:

• Does the applet contribute to learning mathematics?
• Is the mathematics correct and worthwhile?

Interface:

• Usability: Is the visual appearance clear and inviting? Can the user get started quickly? Are the controls easy to use and navigate? Are the directions clear and to the point?
• Use of Technology: Does the mathlet take appropriate advantage of technology, e.g., animation or opportunities for interaction? Does text come in small sections with links to more details when appropriate? Would it be possible to demonstrate the same content equally well using static graphics and text?

Content:

• Focus: Is the mathematics worthwhile and handled at an appropriate level? Does it try to do too much? (Some applications may be good as modules but too extensive to be considered a "Mathlet".)
• Clarity: Is the activity clearly presented? Will students quickly grasp how to get started?
• Robustness: Will the software correctly handle the potential range of inputs? Were there any performance problems?

Technical questions we would like reviewers to answer for good applets:

• Context: Does the mathlet come with sufficient context to be understandable, or does it require the instructor to provide it?
• Suitability: What would be the best uses of this mathlet? Classroom demonstration? Homework with (or without) classroom demonstration? Independent student use ahead of class?
• Plug-ins: Are any plug-ins required? (You may not be sure, but answer as best you can.)
• Platform: What platform(s) and browsers did you use to run the mathlet? Did you encounter any differences?
• Troubleshooting: Did you encounter any technical difficulties? How did you surmount them?
• Cataloging: Is the mathlet cataloged completely and correctly under the "JOMA Journal" heading?
• Redeeming Features: If something strikes you about a mathlet that indicates a second look is in order, even for one clearly unsuitable for JOMA, please put a note in the "Notes" section and the "Keyword" "REVISIT". For example, a site that would be better reviewed as a module, one containing an unusual feature that you'd like to see used more widely, or one where your review would differ substantially from the current one.

Making Better Mathlets

The following selection of comments from reviewers (with all specific names omitted) indicates some of the concerns that kept sites from being considered publishable in JOMA as mathlets. Keep in mind that the authors may have quite different purposes for their work than we did. All of the sites to which the comments below allude were harvested by students without having been submitted for publication in JOMA.

Many mathlets came in large collections, but reviewers often liked only a subset. This may reflect the inability of some websites to be taken out of the context of their collection, site, or course.

Not everything the students came up with was really a mathlet. Some sites' applets were not designed as teaching tools. Others replicate material that can be presented just as well in books, but the authors may have needed everything about the course to be available online.

The reviewers' comments below appear in black, underneath my summary headings in blue.

Some tools were too limited in the number or kind of inputs they could handle. Some were animations that allowed little or no room for interactivity.

• This is a LiveMath worksheet.... The learner can modify the function in some way, but, as always happens to me with LiveMath, I kept making changes that were not allowable. [An article on LiveMath, including some excellent applets using that technology, is planned for the next issue of JOMA.]
• This displays a movie illustrating the limit of the slopes of secant lines approaching the slope of the tangent. While it is possible to interact with it a little bit, it isn't really general enough to be worthy of inclusion.

Some applets had clunky interfaces.

• [About a direction field applet] This applet seems to work well, but the interface is annoying. There are several different pop-up windows that could be condensed into one (or none!). I've seen other applets that do the same thing better.
• [On several different applets] The repaint on this applet was not handled well. I don't like the layout and lack of discussion. Should have given a few more examples.
• [On a root finder] The graph window is too small and the interface is not intuitive. Zoom in is not the inverse of zoom out (there are different centers). There are no tic marks on the graph. This has some value but is not an exemplary example.

Some applets had insufficient mathematical content.

• [About a differential equation solving applet] This just generates a bunch of numerical data. The output is very confusing because the columns don't line up with the headings.
• This computes the first and second derivative and graphs them. It is mainly a computational tool that is not put into any context.

Some were simply done better by other authors.

• [On a predator-prey applet] The user is able to change some of the parameters in the predator prey model and immediately see the change in the population graphs. Other mathlets we've reviewed have done this in a more interesting way.

Some didn't fall within our definition of mathlet.

• This is not a mathlet. It is a Maple worksheet.
• Should be reviewed as a module.

Some failed to take appropriate advantage of technology.

• [On a derivative calculator] This is another one that can churn out examples of computations using the limit definition of the derivative. Again, it is not a great improvement over a Schaum's outline.
• This is just a tutorial with multiple choice questions. Not really a mathlet.
• There is nothing innovative here. The material in the text book is simply transferred into a webpage. Included is a "dynamic" proof of the fact that absolutely convergent implies convergent. I found this to be ridiculous; the lines of the proof slid across the screen, one at a time. Why is this any different from reading the proof line-by-line from a book? The authors are not making appropriate use of the medium. I don't think this is the sort of material we want to highlight in JOMA.

Other comments:

• [Student] None of the documentation could be read using Netscape 4.7 on a Win 98 PC. [Reviewer] That's because it was in Japanese.
• It's a 1996 Applet -- it's not surprising that it didn't run.