### 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.