What are we learning and where do we go from here?

I'm particularly honored to be invited to address you in light of the quality of the past programs you've put on the Web. In fact, I can illustrate many of my own points using the work and ideas of your members, and will frequently do so. I'll also call into play a fine AMS session about using the Web in teaching, which EXPO member Andy Bennett organized at Kansas State last year on Mathematics Education and the Internet.

The last time I really dug into this topic was in April of 1997, for our local MAA section. My paper is still up, and some of the ideas in it are still current. In addition, many of the links still work and are interesting. My style of presentation will be modeled after that talk: the presentation will be extremely low tech so that I can paint a broad picture without encountering technological distractions and courting technological disasters. Also, I hope to entice you to examine this talk in its html form at http://mathforum.org/articles/webteaching, replete with examples and urls.

The game plan:

- what I take to be your special interests, and a general comment thereupon,
- where the World Wide Web fits in, together with its natural programming environment,
- what's good about the Web,
- what's bad about the Web,
- what we can do about it.
My focus will be on the last point.

## (1) Your Special Interests

It would appear from previous programs that the technologies that interest you the most are

- handheld devices
- computer algebra systems (Derive, Mathematica, Maple)
- other computer programs (spreadsheets, dynamic geometry programs, and the like), and integrating their use into your teaching
- the World Wide Web.
(Fortunately, the programs also show that you're also very interested in teaching issues and in the mathematics!) I suspect your interests resonate well with those members of the mathematics community at large who use technology in their teaching.

I'm sure that many of you are aware that a convergence of these technologies is taking place: handheld devices can run computer algebra systems (and TI has recently purchased Derive), they contain dynamic geometry programs, spreadsheets, and computer operating systems. They're also becoming capable of hooking up to the Web. They grow larger and more computer-like even as computers become smaller and less expensive. In some respects other technologies are converging as well; we'll soon see some examples.

## (2) Where the World Wide Web Fits In

It seems likely to me that these technologies - handheld devices, computer algebra systems, other computer programs - and the Web - will be around for some time, coexisting and interacting, and that all will continue to play important roles in the teaching of mathematics. It also seems to me that the WWW is becoming our major means of communication and will thus be the glue that holds together the technologies for teaching. The Web will also be the context within which one communicates about the other technologies, sharing ideas and information via Web pages and Web-based discussions.

Applets are the WWW's natural programs. They can duplicate computer algebra system capabilities, as well as the abilities of special computer programs. As an extreme case, Sketchpad can now save programs in Java form, "Java Sketchpad." Computer algebra systems are developing more highly refined Web capabilities, but when working with Web pages constructed with a computer algebra system, I always have the feeling of being imprisoned within a large and somewhat inflexible program, one to which hypertext capabilities were added late in its life.

Computer algebra systems can do amazing and powerful things that are essential in some contexts, but their capabilities can be overkill in simple contexts, such as the vast majority of teaching situations. For my own tastes the very flexible hypertext world of the Web, aided by small, flexible applets, is where it's at (or will be when applets can do all we need). Again, I neither desire nor expect that computer algebra systems will go away, but I believe that applets will give a better feel for the mathematics and will be more satisfactory in many teaching contexts.

I realize that there is a tension here between

general purpose programs,which are relatively large, expensive, and hard to learn,and

special-purpose programs,which can be quickly learned, are relatively small, and are inexpensive, but which can be too specialized and may not exist for the purpose you have in mind.There are contexts in which each kind of program is appropriate. For example, computer algebra systems are great for more advanced, technical work, while applets have the potential to make a better impact in elementary teaching contexts.

Java applets have many advantages.

* Users love them.

* They are quick and easy to use.

* They focus users' attention on the point being made.

* The technology is not intrusive.

* They can be visually attractive.

--Frank Wattenberg, KC Mathematics Technology EXPO, October 2, 1998.

## (3) What's good about the Web

Having argued above that the World Wide Web is becoming our major means of communication, I'll now look at where things are with the Web today. First the good, then the bad.

(a) Communication/collaboration/sharing:We communicate and collaborate via email, which gets better as it acquires more Web capabilities. The Internet has given us all the opportunity to shape, contribute, and publish, so that we now share our mathematical ideas, our teaching examples and materials, and our personal backgrounds as never before. The Web gives us access to experts, colleagues, and resources at a distance, and connects across communities (mathematicians, teachers, students, professionals, etc.) You can participate when ready, and interactions can be recorded, built upon, and researched.

- Sci.math newsgroup
- Math Archives
- Math Forum
- The Topology Atlas
- Jeff Miller, Earliest Uses of Mathematical Symbols and Earliest Uses of Some Words of Mathematics
- Russ Rowlett, How Many? A Dictionary of Units of Measurement
- Xah Lee, A Visual Dictionary of Plane Curves, Wallpaper Groups

(b) Publishing:In addition to offering vivid color and hypertext, the Web allows for animated and interactive content. It immediately makes available relevant and current real-world materials, showing connections to math in use. Publication of student work provides motivation and student "ownership" of material. Web pages are automatically subject to public review.

- Journal of Visual Mathematics,
- A geometry paper in thoughtful hypertext
- Some student work, - Itai S.M. Kunonga I: Discovering New Frontiers in Calculus
- Some up-to-date real world data: Hurricanes
- Chance events as reported in daily newspapers and current journals, The Chance Database
- Some lovely ThinkQuest competition winners: http://library.advanced.org/12295/, http://library.advanced.org/16661/, http://library.advanced.org/12170/

(c) Help with teaching:The Web is useful for conveying course materials, homework, homework help, and testing, and for getting students to work together. The Web can aid in managing courses by providing information, syllabi, assignments, and discussions.

- Information on course: syllabi, assignments, special information--see many home pages, for example http://www.math.ksu.edu/math240/, http://www.peru.edu/~kincaid/math240/index.html, http://www.jccc.net/acad/math/main/mrc.htm
- Hold effective discussions outside of class http://www.math.ksu.edu/meati/banchoff.html and http://www.math.brown.edu/~banchoff/ma35/,
- Administer non-multiple choice tests http://www.math.ksu.edu/meati/sapir.html,
- Do and check homework, http://www.math.ksu.edu/meati/gage.html
- Train teaching assistants, http://www.math.ksu.edu/meati/baxter.html.
The Web can even help us understand how students learn - or fail to learn - and can help us design for better teaching. John Anderson at Carnegie-Mellon is using artificial intelligence to design a really noteworthy algebra tutor. In his studies, he sometimes tracks eye movement to see how users approach problems. You can clearly see when a student has misread instructions (or failed to read instructions), and you can sometimes even guess a student's underlying approach to solving a problem.

## (4) What's wrong with the Web

(a) Technical problems:The Web can be slow, glitchy, and unavailable when you need it. Math typography is a problem: math symbols don't work well as gifs because gifs are inflexible and hard to search. In general, the development focus still tends toward technology rather than teaching.

(b) Finding what you want:Search engines generally return too much or too little. High-quality material is hard to find. Interesting material you looked at the other day may no longer be there.

(c) Not everything you or your students need yet exists- neither material nor software tools.

(d) Equity:Not all of us have decent access to the Web in our classrooms or for non-classroom use. Not enough schoolteachers are comfortable with its use.Additional complaints from the audience?

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## (5) What to do about the problems with the Web

(a) Technical problems: The Web can be slow, glitchy, and unavailable when you need it.The Web seems, in general, more rapid than it was a few years ago - but there are moments ... ! I counsel patience, but I note that the math community didn't always have to rely on the kindness of strangers; we were on the cutting edge of the Web revolution until the Geometry Center was closed in August of 1998. The Center had one of the first 300 Web servers in the world, and was on the very first Netscape developers' resources page. As we'll see, their good work is still being felt, but mathematicians current have no group to turn to for solutions to Web problems.

Math typography is a problem: math symbols don't work well as gifs (they are inflexible, and hard to search).Help is on the way, as you heard in Andy Bennett's talk last year: an extension of HTML called MathML is coming that will do everything you'd like and then some (it's smarter than TeX). The major browsers will soon support it. WebEQ, a suite of Java programs for putting math on the Web using MathML, was written by Robert Miner of the Geometry Center. (Robert was also co-chair of the W3 group responsible for MathML).

(b) Finding what you want: Search engines generally return too much or too little.Mathematics to the rescue! There's a charming new book just published by SIAM:

Understanding Search Engines: Mathematical Modeling and Text Retrieval,by Michael W. Berry and Murray Browne of the University of Tennessee, Knoxville. They also have an article in the July/August 1999 issue of Siam News that, among other things, discusses using the QR algorithm in searching. (For reasons you'll have to check the article to find out, it's called "Atlanta Organizers Put Mathematics to Work for the Math Sciences Community.")- Not only that, there's a movement afoot in the Web world to bring more useful descriptions of content to the Web. The math community is in the vanguard, and you will soon be hearing further news of "math metadata" in sessions at the AMS/MAA annual meetings in January, "Putting and Finding Mathematics on the Web," Wednesday, 2:15 - 4:15. If you can't wait and would like to be in on discussions now, check out the Math Metadata discussion hosted by the Math Forum.

- Search engines are getting better, too. For example Google assumes "and" between words, not "or," which is a considerable improvement over other searchers. Moreover, it keeps a local copy of the pages it's catalogued, and you can download this cached page, which somewhat alleviates the difficulty of re-finding material that has moved. And it lets you check the citations to pages. It uses a "complicated mathematical analysis" to estimate the quality and importance of pages it returns. I'm advised by someone at Google to keep an eye on the site; in the next few weeks it will contain some information as to what math is used and how. Finally, most people think of Google for returning pages according to how many other Web pages link to them, so sites that are widely linked come higher in search returns.

High-quality material is hard to find.Have you checked out bookstores and libraries recently? The difference is that you have well-developed techniques and there are well-developed tools for finding your way around these paper places, so you can ignore the chaff as you go for the wheat. Search engine improvement will help the Web. Some sites make a serious effort to point only to high-quality material; for example, we have reviewed all of the 5,700 pages in the Math Forum's searchable and browseable Internet Mathematics Library. Other sites that review material include GEM, the Eisenhower National Clearinghouse, and the UTK Math Archives.

Interesting material you looked at the other day may no longer be there.Again, patience. (Action might also help!) I'm trying to interest a large foundation in establishing a reviewed archive so that good stuff will stay around. And Google's approach would help. Other ideas?

(c) Not everything you or your students need yet exists - neither material nor software tools.We need some of the capabilities of powerful computer algebra systems, but we need these to be captured by applets. We must fill large gaps and build more applets that can be devoted to mathematical tasks and combined to further explore math.

This is something we can all work toward, at least when it comes to developing material, text and/or software. You can put up those special examples you use in your own teaching, those labors of love that help your students but which you're hesitant to publish in the Monthly, Math Magazine, or Math Teacher. These may take the form of text - with nice illustrations, I hope - or maybe Java or other software programs.

The power of the Internet comes from its ability to facilitate the sharing of information. With the aid of the Internet our students can have access not just to our best ideas, but to the best ideas of mathematics instructors around the world. Many mathematics instructors have developed a few favorite activities that they have used in instruction to illustrate important points. If these activities could be made accessible to users of the World Wide Web, they would constitute a vast resource for all students to use.

– Joe Yanik and Chuck Pheatt, Emporia State UniversityI see a lot of this sharing of material already taking place, and I'm frankly concerned about the duplication of effort, envisioning a thousand applets showing secant lines becoming tangent lines. I'm also concerned about the overall lack of quality; the Educational Object Economy site has around 560 math applets, but most are not usable by faculty or students.

QualityHow do we assure the quality of math papers and math education papers that appear, for example, in the Monthly? Our usual coin of the realm, whereby we get credit toward promotion and tenure, is peer review. We need to be able to bring this over into educational technology: it's a nontrivial task to construct a single really good applet or other software program, and those who do good work need to be recognized and rewarded.

For the time being, the very lack of quality control of material on the Web can be used to pedagogical advantage. Fred Rickey at Bowling Green University believes that the most serious question is, "How do you teach students to make judgments about the quality of the information that they find on the Web?" He believes that you need to help students develop a sense of skepticism about whatever they read, and he gives examples showing how he does this in math history, http://www.math.ksu.edu/meati/rickey.html.

I hope you all heard "Beyond Kids in a Candy Store," the talk by Frank Wattenberg, your keynote speaker last year. I heard Frank give a related talk last year and his siren song proved irresistible. He inspired me and last May I submitted a Digital Library proposal to NSF that I hope will help make available the applet resources we need in our teaching. I've received word that they wish to fund me at some level, but must await the new budget, sometime this month.

What I plan to do is the following:

The Math Forum is under contract to publish the Journal of Online Mathematics and its Applications, (JOMA) Ladnor Geissinger, editor, Jerry Porter, project director. It will publish reviewed mathematical "modules," Web-based interactive class material of length suitable for anything from a class session to a semester. If you have such material, and it's suitable for publication, please send it to Ladnor Geissinger, joma@math.unc.edu. You can also contact him if you'd like to be involved in the project.

If funding comes through, JOMA will also publish applets, both free-standing and embedded in text and ready for immediate teaching and learning. The applets will be reviewed and tested. Special attention will be given to developing applets that are reusable and interoperable, meaning, basically, that they can be linked and used together. (I also like "reusable" in the non-computer science sense that one could take an applet and embed it in different text to explain math your own way, or could use the applet in a different context.)

Many good ideas for this project come from my fine collaborators. For example, Alexander Bogomolny, whose column on the MAA website always includes interesting applets, was adamant that we should thoroughly cover the curriculum. He suggested a structure for browsing that would be like the table of contents of a generic text, so that faculty and students would immediately be able to find where they are and what's available.

Our goal, then, will not be to review

allapplets, but to cover the curriculum. Consequently, we will search for good examples in well-represented areas, and then move on. The table of contents browse structure will soon show gaps, so those of you who make applets will see what needs to be done.We have a plan for quickly covering the curriculum from pre-calculus to the mid-undergraduate years, provided we get adequate funding. You can look at the grant proposal and further inquire or offer to participate by sending email to applets@forum.edu.

(d) Equity: Not all of us have decent access to the Web in our classrooms or for non-classroom use.Again, this is something where we can all help. Those of us who have good resources need to make clear how they help us with our teaching and our students with their learning, so that colleges, states, school districts, and schools will feel compelled to offer good resources for their teachers and students.

Those currently at the short end of the stick should ask us fat cats for this information and forward it to their local decision-makers. Those of you who have been using technology should look for opportunities to articulate and demonstrate the value you and your students derive from the new technology. Much of the technology is so flashy and self-evidently good that it's an easy sale.

Not enough school teachers are comfortable with its use.At this time it appears that many institutions have funds for technology but not for helping teachers learn how to use it. Many teachers are becoming familiar with the Web as they are swept along in the tidal wave of home use. This should give them a starting place for using simple technology at school.

## Conclusions

It's pretty clear that 10 years ago no one would have had the foggiest idea that a World Wide Web revolution would sweep through the communication and teaching of mathematics. We are living in a remarkable age. We are learning a great deal about how to make use of these amazing new tools for teaching mathematics. The situation is far from perfect, but many improvements are being made. There are situations where by working together as a community we can harness the revolution for the greater good of mathematics education, in particular in

- developing Web material for teaching,
- dealing with questions of equity.
I hope that you will take advantage of the privilege of living during this revolution by helping to shape it in positive directions. It's both worthwhile and exciting to participate.

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