A
New Palette for Applets: Mathwright Microworlds!
Read
about Mathwright Microworlds in our recent article in the Mathematical
Association of America's Journal
of Online Mathematics and its Applications. You will need the
free Personal MathwrightWeb Control that you can download from our
article there, or just below
Imagine
that you create a mathematical web page. You put careful thought
into its design, and you decorate it with instructional text, pictures,
forms and hyperlinks and whatever other gadgets that HTML provides,
that you find useful to tell your story. You then have a "hypertext"
mathematical story that has the additional important property that
is can be connected by hyperlinks to a vast collection of other
mathematical stories on the World Wide Web. One might say that this
story is now 3-dimensional,
by virtue of these connections.
But
something is missing. Readers cannot do experiments or ask "what
if" questions. The demonstrations and arguments are as static
as they have always been in mathematics texts, perhaps more colorful,
but still static. A student may ask: "Well what does the graph
of an epicycloid look like?" But unless you have provided one,
or a pointer to a page that has one, they will not learn it here.
HTML was not designed to provide the support you want. There is
no "epicycloid grapher" and probably never will be in
HTML. What you want is an additional dimension
(say, a 4th dimension) of interactivity.
So
you say, "I can provide this interactivity with a Java Applet!"
And you go to work. You have read that Java is a very powerful computer
language, and it will certainly be possible to do this. And you
were not misinformed. It is a powerful language. But here, you learn
a sad truth. It is a general-purpose language, not a mathematical
language. If you want it to accept (and understand) input from your
readers like
x =
4(5cos(t) - cos(5t))
y = 4(5sin(t) - sin(5t))
You
must do a fair amount of work to teach
it what such expressions mean. And if you want it to graph the curve,
you must teach it that also.
This "teaching" is, of course, also known as "programming."
But teaching a computer is far more time consuming than teaching
a human being. And if you happen to be teaching a few humans also,
you may well wonder where you will find the time to do both.
Aside from resorting to a "supercalculator" like Maple
or Mathematica, you have few other options. And if you only want
to tell a story on your web page, you may well wonder whether your
readers will "get it" (or even be able to read it) in
a supercalculator environment that is designed to support research,
not learning.
One
of the charming and powerful features of Java is the fact that it
is "object-oriented." This means that Java programmers
may do things more easily than programmers did in the past, because
they can first create a world of conceptually simple objects and
their relations, and can then design their programs by thinking
more abstractly -- at the level of the objects themselves -- than
they would have been able to do in the past.
For
a mathematical story-telling program, the teacher/author would "converse"
with the program at a level of abstraction that presupposed computer
algebra style representation, parsing, simplification and comparison
of algebraic expressions. The system should "understand"
rational, complex and decimal number types, and "know"
how to create functions of one or several variables, either real
or vector valued. One should be able to define vectors, matrices,
sets of numbers or names, differential equations, geometric objects,
and so on, easily and informally.
Of
course, it is also important that
the language make available screen objects
that display these mathematical objects, such as graphing windows
that "know" how to draw graphs of functions and curves,
and that can display sprite animations, or draw and manipulate geometric
objects. There should be windows for displaying mathematical formulas,
exact rational (or decimal) matrices, and even pictures correctly,
as well as database tables (like spreadsheets) for input and output
of data. Along those lines, of course, there should be other input
gadgets such as text fields, sliders, checkboxes, and hotspots.
And
all of these must of course be tied to built-in programs that handle
such tasks as solving equations, differentiating and comparing algebraic
expressions, solving differential equations, inverting matrices,
transforming geometric objects, and so on.
Given
such a palette of objects, at such a level of abstraction, teachers
have been able to create colorful, interesting and engaging interactions
and simulations in Mathwright Microworlds. And they have fun doing
it! To see what a few teachers were able to do, starting from scratch,
please take a look at our listing of Microworlds
that teachers have created. You will doubtless be surprised to know
that they are not programmers, but are "ordinary" teachers
with extraordinary mathematical imagination and storytelling ability.
Now,
let's return to the canvas of our 3-Dimensional HTML page and ask
how the palette is applied to the canvas. It works this way. On
each web page, you may insert a rectangle (that we call the portal)
that blends in with the page, and appears to be a part of it. That
rectangle can have any size and position within the page, but it
really represents an additional dimension.
The
portal is in some ways like an Applet, in that its content is automatically
downloaded once from your web page, then cached on the user machine,
and the content fills the rectangle. But unlike an Applet, this
rectangle can have as many "pages" as you want. So when
the reader finishes a "page" she presses a button, or
clicks a hyperlink, and is taken to another page that is displayed
in the same portal. Readers may move back and forth this way among
the pages, and whenever they return to a page previously visited,
there is no wait for objects to be downloaded, and any structure
they might have created (graphs, tables of data, mathematical text
or matrices, etc.) is there waiting for them!
Now
you may well ask: "Well how do you create this portal?"
That is the interesting thing. The portal itself is created in a
separate computer program called Mathwright32
Author, and then "inserted" bodily in the web page.
In fact, the authoring program will create the web page itself with
the portal in it, if you like, and then you can add the HTML decoration
later.
There
are two parts to creating the portal. On the one hand, there is
the screen design for each page. This is all point-and-click (or
as some say, WYSIWYG: "What you see is what you get")
You choose the page color (from 16 million) or a background picture
that can carry hotspots. And then you simply draw
the various screen objects such as textfields, graph windows, labels,
sliders, hotspots or buttons and so on, on the page. These will
provide the basic input and output or both.
Of
course, the other (essential) part of your work is to "script"
these objects. This determines the dynamics of your Microworld --
its behavior. The language that you use to do this is neither
HTML nor Java. It is a high-level mathematical scripting language
called MathScript. MathScript
"understands" the array of mathematical object types and
their transformations, presentations, and comparisons outlined above.
This is the most important sense in which Microworlds differ from
Applets. Mathematical objects and notations enjoy first-class citizenship
in MathScript. To integrate a function, you simply say: Integrate.
To graph a function, you simply say: Graph. And to define a function,
you write something like: f(x,y) := (x+sin(x-y))^2. It is similar
for other mathematical objects such as vectors and matrices, polygons,
ellipses, etc.
Now
of course, the scripts for screen objects are generally not such
simple commands. They can be as rich as you like, with all of the
usual control structures (loops, conditionals, recursions, and so
on). And this means that the only real limits on the expressive
power and range of your story will be your imagination. We now have
roughly 90 Microworlds in the Library.
Why not look them over to get an idea what you can do?
The
Mathwright32 Author program is the only WYSIWYG mathematical Microworld
builder available to teachers today. It was created by teachers,
and its design is based on 18 years of development and refinement
following feedback from the teachers who use it. You may download
the free Author Tutorial (140 pages)
and see for yourself how easy it is now to create rich and powerful
mathematical Applets for your own website, using a point-and-click
interface and a mathematics language that are together much easier
than Java. If you would like to purchase one, you may do that at
the Library Store. You may also download
and distribute to the visitors to your website the free Personal
MathwrightWeb ActiveX Control from the Library Store.
