WebMath: mathematical exposition

[James White]

Dear June,

        The New Mathwright Library http://www.mathwright.com may have what
you are looking for.  It was recently very favorably reviewed in MAA's Focus
Magazine (Dec. '99) and in the College Mathematics Journal (Nov. '99) and it
is a free Library that contains more than 150 interactive "WorkBooks" that
cover a range of Mathematical topics from arithmetic to differential
geometry.  I am the author of the language/platform in which these WorkBooks
were written and are delivered, and the WorkBooks themselves are all written
by teachers of Mathematics and Science, including myself.

        I will let the recent reviews of the Library speak for it, adding
only that Mathwright is quite different from anything that I have seen
available on the web.  The reason for that is that it was written (over the
course of fifteen years) precisely to allow teachers to encourage their
students to experiment with and actively to explore the mathematics in the
stories that they create.

It goes far beyond hypertext authoring systems such as Macromedia or
ToolBook because, while it allows authors to use animations, video, audio,
and other multimedia devices, these animations are live, generated in
response to student input and interactive with it.  For that, it contains a
dedicated computer algebra and graphics language (including also Logo, I
might add), as well as an artificial-intelligence based mathematics
scripting language.  It also makes it easy for authors to create displayed
mathematical formulas in  WYSIWYG windows that also display pictures, screen
snapshots, matrices, etc.  These formulas may be edited by students.  They
may of course also be generated by scripts as the result of student input.
And Mathwright goes beyond dedicated "supercalculators" such as Mathematica,
Maple, or MathCad because it takes the user interface seriously, that is, it
provides an environment in which students can focus on the mathematical
content, the story.

I will include below a brief synoptic overview of the Library, taken from
one of its pages, below, and I invite you and your readers to visit.


Sincerely,

James E. White, Ph.D.

On the Mathwright Library

       This Library is an experiment in computer-based pedagogy.  Its aim is
to invite students to come  into the world of mathematics and science
through structured microworlds (called WorkBooks) that will allow them to
ask their own questions, to read at their own pace, and to experiment and to
play with those topics that interest them.   Teachers may guide these
explorations, but they will not really be able to control them.   Each
Mathwright WorkBook attempts to place the player, the learner, in the
driver's seat.  All WorkBooks have been created by teachers at the
Undergraduate and Secondary levels, with the exception of a few in the Young
Players category that were created by students.
In order to encourage active exploration, the WorkBooks are designed (to the
limits of our ability) to be easy for students to use, but at the same time,
to be truly interactive.   The computational environment recedes into the
background so that the mathematical or scientific topic of interest comes
under the spotlight.  This means that students may often ask questions or
explore ideas that even the author of the WorkBook did not anticipate.  It
requires an expressive and flexible reading environment.  So we place that
environment, an object-oriented computer algebra and graphics language, on
the player's machine.  It is called the Mathwright Library Player.
The Mathwright Library uses the web, not principally as a repository of
"information," but as a source of "experiences."  Thus, the Library is the
web conduit to a collection of interactive explorations that are brought to
the Player's machine.  Once on the Player's machine, each WorkBook has full
access to the language that will bring it to life.
While there has been much speculation about the possibility of delivering
genuinely interactive mathematical content over the web in the form of ...
simple "appliances,"  it seems that the realization of this idea lies far in
the future.  Players create structure as they read Mathwright WorkBooks, and
that structure requires the full services of a complete language if the
players will be allowed to transform and compare what they build, view it
from different viewpoints (i.e. algebraic and geometric as an example) and
to ask their own questions.
Also, players read at their own pace.  Genuine learning requires time and
reflection. It is much more natural for them to print the documentation,
read it, and then to experiment and play on their own (or on lab) machines,
rather than to be required to log in to the internet each time they want to
explore another page in a WorkBook.
Library WorkBooks are hypertext documents.  They vary in size from 1 page to
25 pages, and more.  Many of them look and feel like web pages, and that
fact leverages the experience that many already have with the web.  But in
our WorkBooks, students will be able to do very interesting and exciting
things.
They may place a satellite into geosynchronous orbit, launch a space
shuttle, create an airline routing system and ask it to find routes
satisfying conditions they set, or ask for step-by-step explanations as the
program solves an equation they create. They may, in fact, teach the program
how to implement new functional identities (such as the Pythagorean
identity) to simplify an expression or to solve an equation.  They might
define a differential operator such as the Laplacian, and then draw the
level sets of the Laplacian applied to functions of their choosing.  Or they
might define a "curl" operator and draw integral curves of the curl of a
vectorfield they choose. Such performances are far beyond the reach of
"appliances."
And while some of these examples may appear unlikely, it is a fact that an
author may choose to lead her students in exciting and perspicuous
directions.  An experienced author may create most of the interactions
described above in a few minutes using the built-in language tools!  The
"space shuttle" or "satellite" simulations might take a few hours, depending
on the degree of realism required (See the Gravitation WorkBook for the
spacecraft examples, the Discrete Mathematics 2 WorkBook for the Airline
Routing example, or the Expert Systems WorkBook for the Functional
Identities example.)
Another important fact about authoring in Mathwright is that all Library
WorkBooks are available to be modified and extended if an author wants to
improve one.  Mathwright Author 2000 allows authors to "cut and paste" whole
pages of a WorkBook, or individual objects and scripts.  Many authors base
their own new WorkBooks on existing Library books, cutting and pasting
objects and  pages as they see fit.  This is OK.  The authors of Library
books provide their WorkBooks as "clipmath" and give their permission to
other authors to modify those books if they desire.  Of course, courtesy
dictates that authors acknowledge their sources. Thus, the Library is an
evolving and dynamic entity, not a static one. The WorkBooks that authors
create with Mathwright Author 2000 are immediately distributable to their
students since they can be read with the Mathwright Library Player, which is
freely downloadable from the Library site.
We have also attempted to make it easy for teachers or students to find what
they want in the Stacks.  One may search the stacks in a variety of ways to
locate items of interest. The Library is continually growing, and while it
presently has holdings in a wide range of topics, we hope in coming years to
enrich and enlarge these holdings to fill out many of the lacunae that still
exist.  For that, we will depend on teacher/authors to build ever more
exciting and engaging WorkBooks.





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