In addition to teaching the traditional courses found in a mathematics curriculum, the Classroom is also being used to pilot-test two NSF- supported calculus projects: the "lean & lively" Calculus course currently under development at Oregon State University and the Purdue Calculus Project.

Teaching in this atmosphere provides a fresh approach to the study of mathematics by incorporating personal computers and sophisticated Computer Algebra Systems (CASs) into the curriculum, thereby enhancing a student's understanding of the fundamental concepts of the discipline through the exploration of complex examples and applications.

Some see computers as transforming education the way they transformed industry: teaching more students more quickly and more efficiently. Although this is certainly possible, our experience so far indicates that it is better to think of the computer as "Doing something different" rather than "Doing the same thing faster." The intent is to realize the vision of a "lean and livelier" approach to the teaching of mathematics by taking appropriate advantage of computers and CASs to allow students and instructors unprecedented access to numeric, symbolic, and graphic tools that are usually difficult to get hold of in the classroom; thereby changing the current manner by which mathematics is taught.

The best words to describe our Computer Classroom experience are "interaction" and "collaboration". The course is taught in its entirety in this Classroom and often combines lectures (utilizing the blackboard) and computer demonstrations by the instructor with individual and/or collaborative in-class activities by the students. This inevitably leads to spirited interaction between instructor and student. The spontaneous and simultaneous nature of this interaction is exciting and mathematically stimulating to students. Quite often a student will lean over to another student's computer to point out a syntax error or to make a helpful observation; thereby changing a frustrating experience into a positive one. Recent research indicates that collaborative learning is extremely effective. We encourage collaboration at all times. The reward is the realization that weaker students, who might ordinarily drop the course, become engaged by the stronger students and respond by completing the material, while the stronger students are able to advance beyond the current level of the material through the spirit of exploration that until now was virtually inaccessible in the classroom.

An important concept in mathematics is that of "reversing" a function process. That is, beginning with a function's output, attempt to recover the original input. This is a difficult concept for a beginning calculus student to grasp because it is too difficult to illustrate on the blackboard in any way other than an elementary one. The difficulty lies in having to solve a non-trivial algebraic equation. I usually avoid this difficulty by presenting simple examples which generally leave students unimpressed and uninformed as to the true importance of the concept. This semester, however, I was able to take advantage of the powerful CASs by using a cubic to illustrate the concept. After "looking" at its graph on the monitor and observing that an inverse function existed the class was asked to solve for the inverse function. This required them to solve a cubic equation, which no one knew how to do. Since no one offered a solution, I suggested we turn to the CAS for help. Lo and behold, it gave us three solutions, two of which we quickly discarded because they were complex (conjugate) solutions. (By the way, an important by-product of this was the student's reenforcement of the Fundamental Theorem of Algebra from High School.) By plotting the resulting solution along with the original function on the computer, the students were now able to "see" and understand the concept of an inverse in a non-trivial manner.

Students use the computer to verify and expand on statements made by the instructor at the blackboard. For example, in a course called Basic Analysis, students come face-to-face with difficult mathematical concepts: continuity, differentiability, integration, etc. When they study the derivative, it is valuable for them to "see" that being differentiable at a point is a local phenomenon. The computer presents an ideal way for them to learn this. By zooming in closer and closer in a vicinity of the point in question, the students see the local behavior on the screen. In this class, I used a common example from the literature which involved an expression depending on a variable x. By using the CAS's plotting routines the students were able to zoom in close to the point in question and look at the results on the screen. I noticed that one student's screen was different. Indeed, this student had replaced every occurrence of the variable x by the square of that variable. That was remarkable. I casually remarked that his screen looked different and wondered whether he replaced the variable x by the square of that variable. The student, not sure if he did something wrong, was slow to respond, but eventually admitted doing precisely that. At which point I exclaimed (loud enough to be heard by every student) that that was great! Not only did he sit up straight, but was proud to let straining necks see his display on the screen. I dare say that this student will continue to explore; not only in mathematics but in every discipline that he studies. If only all students learned to do this!

The Classroom is designed so that no computer is physically situated between the instructor and the student. Moreover, each screen is visible to the instructor from the front of the room. The students sit on (comfortable) swivel chairs that permit easy movement between work area and computer area. When the student is at his/her work area, he/she is turned toward the instructor. Most of the computers are placed along the perimeter of the room in the shape of a wide "U" with its open end towards the front of the room. Inside this "U" is a wide "E" with its open end also towards the front of the room. The instructors' computer is at the end of the middle leg of the wide "E". This configuration allows the instructor to see at a glance every student's response to a question involving the computer. More importantly, it forces that alluring screen out of sight when blackboard work and note taking or other written work is necessary.

Our results so far exceeded our expectations, and we plan to employ this technology more aggressively in upcoming semesters at all levels of mathematics. Our surprise at the room's apparent effectiveness underscores the need to expose instructors to this environment; it is literally a case of needing to see it to believe it.