I have been teaching fractal geometry as an introduction to scientific thinking for non-science students since 1986. That year, before computer projection systems were common, I generated the necessary graphics and made transparencies. Since then, support meterial has developed to include live experiments and projection of increasingly complex graphics. These images are a central part of the course, and yet fractal geometry grows so rapidly that the need to incorporate new material defeats the usefulness of static media such as books. (Nevertheless, we do still write books. Peak and Frame is an attempt at a text for this audience.) With class sizes of about 170, daily distribution of 10 or so pages of images to be projected in class is prohibitively expensive. The alternative appeared to be showing the images in class, and have the students try to take notes. This was far from ideal.
Faced with this problem, and given my woeful underestimation of the time needed to make webpages, in the fall 2000, semester I presented a web-based version of this course. My first thought was to have the webpages just copies of the images I would show in class. But then, without the accompanying comments, the message of some of the pictures might be lost. So I prepared text to supplement the pictures, and eventually wound up with a foundation for an asynchronous distance-learning version of the course. Here I will present a sample of some of the pages, and point out a few features of web-based instruction that surprised me.
Of all subjects I know, fractal geometry is the best-suited for web-based instruction. The need for students to manipulate graphic images easily, the centrality of computation for involving students in experiments in the field, and the ease of exploring interdisciplinary projects over the web, are three features of fractal geometry that support this claim. Of course, I am happy to hear from anyone who thinks another field is better. In web-based instruction, we cannot have too many examples.
As a final introductory comment about the differences between web pages and printed pages, I believe the hierarchical organization of material, so easy to achieve with web pages, is a far superior structure for mathematics exposition. A first page gives the overview: the reader knows the main topics to be covered, the orgnization of ideas is transparent. Detailed information is linked to this first page, and those links have their own links to definitions, examples, applications, and connections with other pages. The reader can sculpt the particular text he or she desires. Already familiar with the definitions? You need not bother skipping over those parts of the text: simply don't click the links between word and definition. You're more interested in proofs than examples? Just don't click the link to the examples. Huge amounts of information can lie beneath a single link, so you don't have to search for where the examples end and the proof begins. Moreover, the author can link each part of a proof to an illustrative example. Those needing more examples can click those links when needed; those preferring an unimpeded proof need see only that. Half way through a proof you need reminding of a definition? Click the defintion link of the word. These are just a few examples of the flexibility of hierarchical organization; there are many more possibilities.
Years ago, at the AMS Centennial meeting I remember a wonderful talk by Joe Harris. After introducing a new definition, the next few times he used that term, Harris reminded the audience of the definition. That was such a simple act, but made the lecture much easier to follow by those of us unfmailiar with the details of Harris' field. Now by inserting a few links, every web page can remind readers of a definition, if they need reminding.
Converting this web page to linear text for the printed conference proceedings was a sobering task, and helped focus my thinking about printed pages versus web pages, or more generally, about text versus hypertext. Hypertext is so much more flexible, not just by including animations and links to software, but by allowing a multitude of organizational schemes within a single document. As much as is possible, I am going to present all my pedagogical writing as web pages or hypertext. This flexibility is not possible in print, but done right, makes a much more effective teaching medium.
Next I present a few examples of web-based course design. Keeping with the theme of the conference, I focus on Iterated Function Systems. Click on the picture to go to the corresponding section.
| What are fractals? |
| How do we grow fractals? |
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| Symbol-Driven IFS |
| Data-Driven IFS |
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This is the simplest of beginnings. The real power of web-based instruction is not yet even imagined, because most of us grew up as students before the web was common (or even before the appearance of personal computers, for some of us dinosaurs). The next generation will get it right, but we have to start with something. These are a few of my first thoughts.