The JOMA Applet Project: Applet Support for the Undergraduate Mathematics Curriculum

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Main Proposal

1. Audience

This project will produce a digital library of small platform-independent electronic tools for teaching mathematics. The audience is mathematics faculty, students, and developers. The grant is specifically targeted at college-level material, but because many mathematical ideas span the high school and college curriculum, this set of resources may also be useful to high school teachers and students, as well. We anticipate eventually broadening the project to include pre-college levels. While the current grant will not allow time to collect such material, its design allows for expansion in the future.

In addition to addressing materials specifically designed for the undergraduate mathematics curriculum, we will make a special effort to enable interdisciplinary use of mathematical applets by cataloging those in science, engineering and technology disciplines and constructing bridges to other Digital Library projects. Together we will build federated search engines, develop consistent metadata, and share approaches and materials.

2. Why this is important

Our digital library will

  • provide access to powerful mathematical ideas,
  • encourage developers and focus their work,
  • model effective use of applets and tools,
  • facilitate resource building (applet interoperability),
  • provide curricular structure for applets,
  • support coherence and completeness of technology for the undergraduate mathematics curriculum.

One cannot overemphasize the importance of the tremendous surge in applet construction now taking place. The reader is invited to search for "XX applet" where XX is any elementary mathematics topic. A search will yield many applets for Newton's Method, for Gaussian elimination, power series, or Euler's method -- for almost anything in the undergraduate mathematics curriculum. The Math Forum search engine [11], arguably the best for mathematics education and elementary mathematics, has over 400 annotated applets in its repertoire but the Forum has lacked the resources to make a concerted effort to find, review, and catalog others.

Small interactive programs such as applets are fundamental to using the World Wide Web for learning -- they add interactive images, tables, and computational power, and can offer surprising and wonderful teaching possibilities. A site that should convince any college-level mathematics teacher of this fact is Alexander Bogomolny's Cut-the-Knot column, which appears each month on the front page of the Mathematical Association of America's MAA Online [12]. There is much other wonderful work, but the general quality is uneven, it is widely scattered, and good material is hard to access for immediate use.

Java applets are important because they

  • are extremely easy to use,
  • can run on any platform,
  • cost little or nothing,
  • can come embedded in a lucid discussion of mathematics,
  • have captured the imagination of many faculty as a means of presenting mathematical ideas to their students.

Several very fine computer algebra systems (Maple, Mathematica, etc.) have Web components. These can produce useful pedagogical material of the type we envision, and such tools are currently essential for some advanced work. However, they are large, expensive, require considerable training for most people to use effectively, and in general do not integrate well into Web pages. They are supported by large commercial enterprises which we will approach about possible cooperation, such as sharing of code libraries. Elsewhere considerable energy and talent are being invested in small, easy-to-use, special-purpose applets, and this energy should be effectively coordinated and leveraged for the betterment of mathematics education.

Despite all the energy being put into applet development, curriculum coverage is as uneven as the quality of the applets themselves. A recent search has led us to estimate that there are conservatively 30 applets for numerical integration and 60 for least squares. Some applets (such as those for function graphing which exist in vast numbers) are general purpose and can stand alone.

Applets are likely to become even more important in the future since there are already some applet-generating tools, such as Geometer's Sketchpad: after first constructing a sketch with that program, there is a Sketchpad-to-Applet converter [13]. In addition, there are movements afoot to develop interoperability standards that will make applets basic building blocks that will operate well together (Java Beans [14], Remote Method Invocation [15], etc.), apparently supported by the Digital library Initiative. The Advanced Distributed Learning working group on Shareable Courseware Objects is slated to introduce draft standards in the near future. Our project will advance these directions in applet construction while developing a fully functional and immediately usable digital library.

3. What will be collected

We will assemble a digital library of small platform-independent electronic tools for teaching mathematics. At this time we imagine that the majority will be Java applets, but this is subject to change in the evolving world of the Web. We will collect programs that are electronic, interactive, go beyond text, and insofar as possible are platform independent (special gems may be included for inspiration even if they are not). For convenience, most of the following discussion will be couched in terms of applets because they are the most visible of the materials that need to be collected, but we will be alert to the development of other electronic resources and will collect them as well. For example, chemist Arthur Ellis of the University of Wisconsin has produced diffraction gratings that can be sent over the Internet, printed on an overhead transparency, and used as manipulatives. Other examples include video clips marked to show the trajectory of a moving object, such as a bouncing ball [16].

To make them immediately useful to students and to assist mathematics faculty, applets, diffraction gratings and the like should be embedded in text. We will call this pedagogical pairing "teaching units." We will also look for teaching units -- expository text with embedded applets, ready for teachers to incorporate and students to investigate. The teaching units will also be subject to review.

Our ideal basic applet will come already clothed in a teaching unit, or even as the basis of several units. The digital library will contain screen shots for rapid skimming, and sometimes other ancillary material such as streaming video showing student use.

4. Structuring the work

Our work will be structured around the undergraduate mathematics curriculum. In the first year we will concentrate on basic courses, defined to be pre-calculus, single-variable calculus, and elementary statistics. Later we will move on to linear algebra, differential equations, several-variable calculus, discrete mathematics, geometry, number theory, and so forth.

We will construct generic course outlines for the courses under consideration containing the usual topics covered -- esentially a tree with some crosslinks. There will be occasional crosslinks between nodes in different trees, such as for least squares, which can be approached in a number of different courses. Upon these tables-of-contents trees we will hang the applets and teaching material, an approach that will help us focus on covering the curriculum well rather than exhaustively testing all secant-to-tangent-line applets.

These tables of contents will be presented to teachers, students, and developers as a means of browsing the JOMA collection. This approach will give a familiar context in which to look for related material. The content trees will also be used to secure the help of faculty and developers in filling curricular gaps, and will offer a framework for developing curricular connections.

5. Searching, Reviewing, Testing

Guided by the course contents, search teams will find relevant applets on the Web. There will be four teams located at four different colleges or universities, each comprised of a faculty member and from three to five students. These teams will classify the applets as"yes," "maybe," or "no" (and if "no," why). Using Web forms they will then enter information about the applets into the EOE and JOMA databases. The teams will have different foci, changing with the year, but with one team always specializing in applications of mathematics.

Each summer we will hold two workshops for the reviewing teams. The first summer, one will be devoted primarily to the pre-calculus and calculus curricula, while the other will first focus on elementary statistics, with appropriate variations in subsequent summers. Each workshop will consist of six faculty and will meet for five days. It is not expected that they will be able to look at all the promising material, so the tables of contents will be used to guide the work: after a topic has been reasonably covered, remaining material will be held until time permits. Teams will first work from the preliminary "yes" and "maybe" lists, so that a decent curricular base of quality material will be quickly established.

There will be an applet testing service at St. Olaf College. A group of faculty and students will work with a Math Forum staff member to develop a simple testing suite that checks performance important to users. Each published applet will then be tested for basic mathematical and other functionality, and ease of use. Useful information, e.g. the versions of the browsers under which it was tested, will be clearly presented with the applet.

6. Maintenance issues

In the rapidly changing world of the Web it will be necessary to enlist user and developer support to provide appropriate software maintenance. Consequently, each library entry will come with a link to an archive of user comments and discussion. In this way users can report bugs and problems with new browsers and it will be possible for programmers to announce fixes and new versions well before the journal is able to review them. Search features will allow users to track bugs and find fixes and updates expeditiously, which should provide motivation and useful feedback to developers. Faculty will be able to upload their own versions of teaching materials that incorporate applets.

We will attempt to focus on ideas, which do not go out of date, rather than on software, which is destined to lead an all too finite life. Teaching units can be preserved while their applets are replaced as they become obsolete or are supplanted, but it will be impossible for JOMA (or any other organization) to assume responsibility for updating programs.

7. Sustainability

We will aggressively publicize the digital library through articles in professional journals and presentations at conferences, both national and local. Awards for applets and teaching units will likewise be publicized. We will work with textbook publishers to make users aware of this new support material for the mathematics curriculum. After the initial phase of developing and publicizing the collection, we expect that a critical mass will be attained so that future developers will be aware of the project and will submit their work to JOMA. In this way, the life of the collection will be linked to that of an ongoing electronic journal. We also plan to develop the communities of faculty, students, and developers into self-sustaining entities.

We expect modest ongoing costs for publishing applets; however, after three years JOMA will be a professional publication, staffed largely by volunteers, and motivated by professional obligation. There are likely to be costs for the continued software testing, annual commissioned survey articles on what is available in the collection, released time for the Applets Editor, and the like. We expect to meet these costs through commercial sponsorship and sources such as fees paid by publishers for links from the JOMA collection to the table of contents of their publications.

Discussions with commercial publishers indicate that they would be amenable to urging their authors to construct packets of applets that can be indexed to appropriate sections of conventional textbooks. This information could be supplied to adopters, actual or potential, and faculty materials could discuss the utility of such an electronic packet within the context of the specific table of contents or course outline. Furthermore, publishers should be willing to pay for this supplementary asset, since it would incrementally increase the value of their basic texts.

We will work with other related science, engineering, and technology projects, with JOMA, and with the Math Forum to discover appropriate sources of revenue to cover the needs of the library and its user communities.

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