I describe in this paper a new type of web document -- the Mathwright Microworld -- that appeared recently at the New Mathwright Library and Café. I will discuss the Mathwright Microworld from the points of view of both

• its readers -- typically students of mathematics -- and
• its prospective authors --typically teachers of mathematics.

A Mathwright Microworld is an HTML document (i.e., a web page) that an author may create with any handy HTML editor. The document becomes a Mathwright Microworld when the author embeds a "portal" that he or she creates independently with the Mathwright32 Author program. The portal in a Mathwright Microworld is a rectangular region of the web page which is, in some ways, like a Java applet. Its content is automatically downloaded -- just once -- from the author's web site when the reader comes to the page, and it is then cached on the reader's machine. This portal may be designed to blend with the background, so that only its content distinguishes it. But unlike most Java applets, this rectangle can hold as many "story pages" as the author wants to create. These story pages remain in the portal on the web page as the reader enters this new dimension of that page. Thus, the reader who is finished with a story page presses a button or clicks a hyperlink and is taken to another story page that is displayed in the same portal.

Before you can view a Microworld, you must be using Internet Explorer 5.0 or later, and you must download and install the (free) Personal MathwrightWeb ActiveX Control. Please do that now.

In the next section, our first example, Transformations of a Function, illustrates the appearance and behavior of a Microworld. If you would like a more detailed explanation, go to our Definitions of Terms.

Published June, 2002

© 2002 by Bluejay Lispware

Transformations of a Function, written by Mike Pepe of Seattle Central Community College, is a microworld with eight story pages embedded in a single web page. It has a menu from which you can navigate to most pages, or you can move from story page to story page by clicking buttons. In this book, each page allows you to experiment with a certain type of transformation of a function. If you change data in any of the text fields, the changes persist as long as you are in the book. While you experiment in the book, use the menu buttons on the story pages to navigate. When you finish with your exploration, close the window to return here. You will see a pop-up message telling you that Mathwright is stopping. Click OK to proceed. Whenever you leave a microworld page, you will see this message. It means that the microworld session is closed.

You may enter the microworld now by clicking the hyperlink or the image above.

You may move back and forth among the story pages of the microworld, and, whenever you return to a story page previously visited -- even after you have closed and reopened your browser -- you will not have to wait for objects to be downloaded again. Once cached, objects are immediately available on future visits. In fact, during a single session, any structure you might have created -- a graph, a table of data, mathematical text, a matrix, etc. -- will be there waiting for you when you return to a story page. This persistence of data is often useful, as our next two microworld examples will show.

Microworlds can be designed to encourage students to experiment with an idea from different perspectives. They usually allow students to approach a topic from their own levels of understanding and to ask their own questions. This style of storytelling can be both integrative and constructivist, and it can support visualization in a way that stand-alone demonstrations seldom do.

Now we consider another microworld embedded in a single web page, one for which persistence of data is a useful feature. The Epicycloid microworld has just two story pages. It is designed so that readers can compare two very different ways of constructing these curves. We also begin to consider what is involved in building microworlds like these.

Imagine that you are creating a mathematical web page about epicycloids. You put careful thought into its design, and you decorate the page with instructional text, pictures, forms, hyperlinks, and whatever other HTML gadgets you find useful for telling your story. You then have a hypertext mathematical story with the additional important property that it is linked to a vast collection of other mathematical stories on the World Wide Web. You make the observation that a certain geometric construction yields a family of curves parametrized by a and b:

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 "What does the graph of the cycloid look like if I make the a parameter negative instead of positive?" Unless you have provided an example, or a pointer to a page that has one, the student will not learn the answer here. HTML was not designed to provide the support you need. The equations above are only a picture with no "life" or meaning in the page. What you want is an added dimension of interactivity .

So you say, "I can provide this interactivity with a Java applet," and you go to work. You know that Java is a very powerful computer language, so it will certainly be possible to do this. And you were not misinformed: Java is powerful. But you learn a sad truth: Java is a general-purpose language, not a mathematical language.

Creating a Java Applet from scratch that can support exploration with cycloids can be daunting. We could start with the parametric expression for the curve given above. The Applet would have to accept the parameters a and b from the reader -- easy enough! -- and then draw the curve. This is still fairly easy to do in Java. The simple two-page Microworld below is, on its first story page at least, essentially a Java Applet It invites readers to experiment with parameters a and b, and it draws the graphs for them when they press the "Draw the Cycloid" button. Then, if the reader presses the "Animate" button, it shows how such a cycloid can be constructed by rolling one circle on another without slipping, tracing the path of a point on the circumference.

Such things are fairly easy to do with Java applets, but the reason we do not see more of them is that they do take time. One usually has to start from "first principles" to make a simple demonstration like this. Even if one has access to a good class library, it is necessary to tweak and modify the code at a very concrete level in order to get it to work. Such a project might take days -- more likely weeks -- using Java alone.

But the two-page Microworld on this page required only two to three hours to write. It is based on earlier work by Mathwright authors Mike Pepe (Seattle Central Community College) and Samuel Masih (Albany State University). The reasons it was so easy to write:

• The screen design required only a few minutes of drawing. Once that was done, all that remained was to "script" the screen objects -- buttons and pages in this case -- using Mathwright's high level scripting language called MathScript.
• With Mathwight Library's Open Source convention, some of these scripts were already available in earlier workbooks at the Library, so all that was needed was to "tweak" and extend them to work for this microworld.

A virtual learning environment is more useful to the extent that students can ask a wider variety of questions within it. With the appearance of Java-enabled web browsers, it became possible to put small interactions in the form of Java applets within the browser itself.

Mathematical Applets are, however, a somewhat paradoxical solution to the problem of bringing interactive and live mathematical learning environments to students on the web. While applets make use of some of the most advanced and sophisticated technology available to the web, and they harness the power of a truly elegant and powerful language, they have placed the creators of these applets in the prehistoric conditions of the early '80s, where programmers had to build pretty much everything from scratch. You must do a fair amount of work to teach Java the meanings of expressions such as the cycloid formulas or complex polynomials. And if you want Java to graph a curve, you must teach it how to do that also. 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.

So we see a proliferation of dedicated "appliances" on the web, each devoted to the illustration of some particular mathematical point, with very little -- if any -- communication among them. But it is precisely the possibility of communication among tools that can lead to the integrated experiences for students that can elicit new questions and arouse their curiosity. We saw a very simple example of that communication in the epicycloid microworld.

Of course, it is the virtue of Java that programmers can easily develop reusable class libraries that make possible an object-oriented approach to the development of mathematical applets -- building them from components. But very few teachers have the time or the inclination to use Java in this most elegant way to place mathematical explorations on their own web pages. So they (and their programmers) often have to "reinvent the wheel" again and again to do such simple things as parsing input expressions, or drawing graphs.

We will see another pedagogically useful example of this communication in the Heron's Formula Interactive Web Book. A Mathwright Interactive Web Book is a microworld that is distributed over more than one web page. These books may feature internal navigation among the story pages, or they may not. Heron's Formula uses internal navigation to "prepare" an experiment on one story page, and then to "perform" the experiment on the next. Later, we will visit a microworld called Cardano that is distributed across 12 web pages but that does not use internal navigation at all.

Interactive Web Books such as Heron and Cardano have in common the fact that they unfold a dynamic story. In general, the author of such a story will have an easier job if all of the tools needed to support that story are available in one place. In this way, the author can reuse and re-purpose these tools as the story unfolds. While the advantage for the author is obvious, the pedagogical advantage for readers is less obvious. Such Interactive Web Books, built on a coherent and straightforward functional plan, lead to richer experiences for readers. Readers are invited to participate in the story by

• making and testing hypotheses,
• preparing and viewing simulations -- such as the wheel rolling on a wheel, or the ones we'll see in Heron -- and, in general,
• by asking their own questions.

The Interactive Web Book Heron's Formula explores the geometry behind Heron's famous equation for the area of a triangle in terms of its sides. The formula may be understood by asking which quadrilateral with assigned side lengths has the largest area. This book has several experiments embedded in its pages, one of which allows the reader to vary the shape of the quadrilaterals to discover the surprising answer, and thereby, to discover Heron's formula.

Nearly all of the topics discussed will be accessible to a student who is comfortable with algebra and geometry. The crucial step of the argument, however, uses elementary calculus and may, as a surprising application of the ideas of limit and derivative, be taken as a motivation for studying those concepts.

To visit Heron's Formula, click the hyperlink or the image above.

Perhaps the most important difference between the interactions that appear on the pages of the Heron book, from the viewpoint of authorship and web design, is that they were not written directly in Java but were created in a high level, object-oriented mathematics scripting language called MathScript. Further, the visual design was graphical "point-and-click" or "What you see is what you get." This combination, using the Mathwright32 Author™ program, produces efficient Java code but does not require any knowledge of Java itself. It is much simpler to write books with this system than with Java.

Our final example of an Interactive Web Book is Cardano, which develops and illustrates Cardano's Method for solving cubic equations. For that purpose, the book uses some tools that will offer a few experiments with complex algebra and then with abstract algebra. It finishes by demonstrating graphically an interesting property that cubic functions have when all of their roots are real. In that case, the three real roots are the projections of three equally spaced points on a certain circle (centered on the inflection point) to the real axis. This fact is non-intuitive and surprising, and it really benefits from the integration of two tools: One allows the reader to select cubics that satisfy the discriminant condition and to visualize how the shapes of cubic curves depend on the two parameters. The second tool demonstrates the above-mentioned fact, which is very helpful for a geometric understanding of Cardano's method.

The Cardano Interactive Web Book combines a variety of tools, including

• graphing tools,
• an abstract algebra "calculator" that allows the reader to explore some interesting properties of a certain complex algebra,
• an expert system that solves cubic equations (among others) step-by-step, giving explanations for each step, and
• many exercises that combine modern algebra with graphics.

It develops a novel and straightforward approach to Cardano's method, along with a simple means to understand it.

You may visit the Cardano Interactive Web Book by clicking the link or the picture above.