Recent reviews of Mathwright in the November, 1999 issue of the College Mathematics Journal or in the December, 1999 issue of the Mathematics Association of America Publication, Focus have brought us to the attention of a wider audience of mathematicians and educators. In this issue, we discuss some recent developments that point the way to a possible future for research in mathematical pedagogy. We invite you to join us in this adventure with your comments, criticism and advice. You may write to us at : info@mathwright.com
1] Abstract Algebra in your Browser
In earlier Newsletters, we announced Lava! and we posted a number of WorkBooks at our Lava! Page Today, we talk about the structure and the pedagogical purpose of a new Lava! WorkBook called cardano. This Lava! WorkBook is an extension and refinement of an earlier Library WorkBook called Cubic Equations. The WorkBook is 20 pages long, with 7 laboratories, and it comes with a downloadable and printable 42-page manual. Java is no longer the language of “dancing Dukes” and hermeneutic, single-purpose Applets. It never was. With this example, we intend to show a new role for Java in support of mathematical pedagogy.
As with all of the WorkBooks in the Library, cardano aims to invite the reader to ask questions about the mathematical story it develops. And it attempts to invite questions by illuminating those ideas through dynamic and playful illustrations that accompany the story.
The story in cardano develops a method for solving cubic equations in an interesting new way. In order to do that, it relies on a heuristic approach, developing the algebra of "cubic numbers." These "cubic numbers" are just the elements of the complex group algebra generated by the cyclic group of order 3. As it happens, Cardano's method reduces to this: If you are given a cubic equation: P(x) = x^3 + a*x^2+b*x+c = 0, with a, b and c complex, then you may look for a cubic number w such that (in the algebra of cubic numbers) P(w) = 0.
As soon as you write this condition down, you are led directly to Cardano's method for solving cubic equations. This replaces an arcane and seemingly unmotivated construction with a very simple and straightforward procedure, one that is easy to remember and use, and it discloses the interesting role of the cube roots of unity. A later WorkBook, based on an upcoming paper of James White and Dan Kalman, will show how this method may be extended to quartic equations (naturally using "quartic numbers": the complex group algebra generated by the cyclic group of order 4). We discuss in the article why this procedure fails to yield a quintic number for polynomials of degree 5.
Now a WorkBook that invites experimentation with cubic numbers has to be fairly versatile. It must allow the reader to define cubic numbers, and do algebra with them. Thus, along the way, it must provide an exact complex number type, as well as a "cubic number type." Further, there are many natural bridges with geometry and graphics that one wants to provide for the reader in order to develop intuition.
The cardano WorkBook is therefore a mathematical teaching/learning environment that supports exploration in abstract algebra. It covers a range of topics from graphing and factoring polynomials, exact arithmetic and algebra, differentiation, complex and cubic algebra, linear algebra and more. But cardano is a single 20-page WorkBook that plays in Lava! And the Lava! Applet can play an unlimited variety of WorkBooks on a wide range of mathematical topics. This means that Lava! must be a large and versatile Applet. Lava! has its origins in the Piagetian idea that gratuitous play is the surest source of new and enduring knowledge. And so it tries to provide, for each of its stories, a "playground" for the reader. In this light, to the extent that it is possible, it is desirable to allow the reader to ask her own questions. These questions, asked in the spirit of play, are the ones whose answers she is most likely to understand. Computer learning environments have evolved over the years in their flexibility. The aim has always been, and will continue to be, to smooth the user interface, and to create an illusion that brings its story to life. In the same way that Logo was in the '80s, Lava! is an experiment in mathematical pedagogy. However, Lava! makes some requirements (which will soon be anachronistic) on the user machines.
For best effect, the hardware setup should be:
· 1024 x 768 resolution graphics with 16-bit or True Color
· 64 M RAM
· A 56 KBaud modem connection (or faster)
· 450 MHZ Pentium II Processor (or faster)
· In addition, Lava! requires the Java 1.2.2 Plug-in for your browser, and will offer to install it automatically if it detects that you do not have it.
In many ways, despite its ponderous size and its present lack of speed, Java is an ideal language to support Piagetian learning. This is because Java's object-oriented structure makes it easy to generate the flexible run-time environments that can support Piagetian learning. Try it out!
James E. White, Editor
If you came here from the old building,
