WEBCALC I

Dept. of Mathematics

Texas A&M University

College Station, Tx. 77843

The WebCalC Group is constructing a web based first semester calculus course. It has been and is being used at Texas A&M University and a few junior colleges. The on-line course features:

• sound mathematics,
• a CAS engine,
• a multi-layered approach in presenting the material,
• student interactivity,
• randomly generated quizzes which are immediately graded with results reported back to the student,
• excellent graphics,
• color to emphasize various parts of the material.

The URL for our course is

http://www.math.tamu.edu/~webcalc/mindex.tex

In the summer of 1997 a group of four faculty at Texas A&M University decided to create a web-based calculus course. Why calculus? Why no college algebra, pre-calculus, or differential equations? We wanted to begin with a course that could utilize the full capabilities of Maple or Mathematica for symbolic mathematics. We also wanted to guarantee the intended audience would be at a level beyond remediation. There is also much experience among the authors in using technology to teach calculus. A subject which, for some time, has been on the front line of technology assisted mathematics courses.

There are several ways to present materials over the internet. We decided to use Scientific Notebook for the following reasons:

This program allows one to easily introduce mathematical syntax into a document. It has a Maple kernel, which means that one has a real time computational engine. It also functions as a web browser, in the sense that it has the ability to download appropriate files from the web.

Two disadvantages with using Scientific Notebook as a web browser are:

1. It will run on Windows 95, 98 or NT systems only.
2. To be able to access all of the features of Scientific Notebook requires the full program, not just a free viewer. Retail cost is about $75. This additional cost to students is in contrast with the two most popular browsers which are freely available.

At present, when we teach using WebCalC, our students are in front of a terminal three hours per week, and are instructed to read certain sections. The instructors walk around the room answering questions as they arise. We also saw our students one additional hour per week. This fourth hour was used as a recitation/problem solving session. Three major exams were given during the semester. The exams were essentially duplicates of exams other first semester calculus courses were giving, and our student's grades were comparable to the grades given in the other sections.

It is essential that class time remain focused. Weekly goals must be made and attained. Students need to have their noses kept to the webstone. Students cannot be allowed to drift. One technique, which helps to insure student progress, is to administer lots of quizzes, both in class and take home.

• A perhaps unexpected benefit of this method is that students are a lot more proactive in learning the material. Passivity gets them nowhere. They have to, at a bare minimum, click to see what comes next. It is extremely encouraging to see students actively engaged in educating themselves. Our better students help us improve the course. During the summer of 1998 a small group of high school students (mostly juniors) from several neighboring high schools spent about 5 weeks studying calculus with WebCalC. The feed back we received from them was for the most part positive. One student stopped coming to the lab, as she and her family were traveling. A week or so after the course finished we received an e-mail message from her. She had apparently finished the material at home on her own, and then sent us a fairly long description of errors and suggested improvements.

• When we teach calculus there are several levels of expertise we expect our students to attain. The first and most basic is an understanding of how to compute various items. The next level is a basic understanding of the more important definitions. A third level is the ability to understand and construct proofs. With a textbook every student, no matter what their abilities and needs, will be exposed to all of these levels and more. In a web based course this need not be the case. In fact this is one of the features of our course that we are excited about. We have built the course in layers. The top layer consists of material that every student must learn. For example, the statement of the theorem on how to compute the limit of a sum in a textbook is followed with its proof. In our course this is not the case. If a student wants to see the proof, they click on a link to a page which has the proof. We are also able to encourage students to work through examples themselves. A demonstration of this is shown below. When you see a phrase similar to "what next" imagine this is the end of what the student sees, and the rest of the calculation is not visible until the "what next" link is clicked. Although the links appear to be active, in this document they lead to nonexistent pages.

Compute by using the definition of derivative if .

• One last advantage of this medium, it is dynamic; not static as the traditional text book. This means that errors can be corrected, and expository material added or deleted almost immediately.

There are numerous missteps in a project such as this, and we have made many of them. A few of the more serious ones are:

• Not deciding early in the project the format of the web pages. This has cost us much time and effort. In our defense we are developing something for which there is little precedent, and the fundamental laws upon which to base this edifice are not yet known.
• Over estimating the ability of students. By this we mean that the majority of our audience is not trained to learn on their own. They have little practice in reasoning their way through an abstract argument or any argument for that matter. Mathematics is a foreign language to them. Examples must be complete with few if any steps left out. Answers to questions should not be given in a simplified form without demonstrating how that form was attained.
• Avoid, at all costs, the temptation to not leave anything out. It is very easy for the pages to become cluttered with too many unnecessary words, details, and minutia.
• We are so used to the serial ordering of material in a textbook that we rarely give thought to the problem of finding a particular example or theorem or whatever. We remember the approximate location of the material and can easily find it. With web pages it is much more difficult. The material is not presented serially. We are faced with the equivalent of trying to find something in a collection of books, and we're not even sure which book to look through. Students have trouble finding specific items, and so do the authors. Topography is of paramount concern, and we have not yet satisfactorily resolved this issue.

We think the following is a good first approximation of the "Rules for Web-Page Design"

• Web pages should have as little text as possible.
• A judicious use of color for emphasis and/or to highlight an important idea.
• Entertaining and informative graphics.
• Cul-de-sacs

The idea of a cul-de-sac is simple, and is a design device to help our students maintain a sense of the location of the material they are reading. For example, at the end of a web page, we may have hot buttons that are links to some additional examples. If you go to one of these examples you can link to any of the additional examples, or you can return to the originating page. You cannot link to further material directly. The following graphic illustrates this.

In summary, we have found that students learning calculus through WebCalC do very well. They quite often express the feeling that they are learning the material better than they would in the traditional lecture/recitation format, and credit this to the realization that they are being proactive in their education. They are actually having to (horrors) read material and think about it.