Despite these concerns, I am a firm advocate of the judicious use of computers in the classroom. Indeed, I have used the electronic classrooms nearly every term since the first one opened in the basement of the Humanities building, and have purchased projection equipment for our department. Courses where I have made significant use of computers include
Math 127 ,Math 53 ,Math 15 andMath 19 . The first of these, a course on numerical methods which deals with the issues involved in using computers to perform mathematical work, is a natural one to incorporate computer demonstrations. In it, I use programs to illustrate algorithms graphically (e.g., Newton's method closing in on a root of a polynomial), and also to experiment with the processes we develop to determine their weaknesses (e.g., how does a small error in the initial measurement of a value propagate within an algorithm?).The second course, "Visualizing the Fourth Dimension", makes extensive use of computergenerated images and movies to give the students insight into fourdimensional phenomina. The "Selected Course Notes" section of the
Math 53 web site includes dozens of animations that I produced using theStageTools
package that I developed. These are used in class to motivate and illustrate the material, and are available to the students outside of class for their own study. Indeed, these materials have been used by at least one other course at another institution.I am careful, however, about how I use computers in the classroom, and I only use them when I have a clear pedagogical goal in mind  one that I can't achieve more readily at the board. For example, in a multivariable calculus course it is far easier to draw the graphs of functions on screen than on the board (since they are threedimensional), and they can be rotated and manipulated in other ways that simply are not possible at the board. On the other hand, for singlevariable functions, the board is perfectly adequate for most situations, so I use it only when a computer presentation is required, or where many computations need to be performed. I do not, for example, see much value in using
PowerPoint
to present my class lectures. Although they may be more readable than my writing, and one can print them and hand them out to the students or put them up on the web, I don't believe that this adds significantly to the students' understanding. Indeed, it is my experience that the act of writing in ones notes (and the act of selecting what is important to write) is itself valuable to the learning process.Of course, we have been giving blackboardstyle lectures for thousands of years, so the things that work in this medium have been fairly well worked out over that time. In fact, the examples (and the curriculum itself) for most math courses have been chosen precisely because they work at the board. Simply moving these examples onto a computer, or adding a few computer exercises to such a course, is not an effective use of computers in the classroom. For example, damped harmonic motion (or "the springmass problem") is a classic example in differential equations; but one reason for this is that it is tractable enough to be done at a blackboard, including its graphics. I have seen demonstrations using
Mathematica
that show highly rendered masses bobbing on physically accurate springs; but these give no more understanding than watching a teacher move his hand up and down next to a sketch of a spring on the board. Even when the motion of the spring is tied to the graph of its position this is only marginally better than what can be done by hand.Some uses of the computer are valuable, and others clearly are not. The goal is to find the problems that are appropriately treated by computer demonstration. Computers are relatively new (compared to blackboard lectures), so it will take some time (perhaps decades) to find the right idioms for their use and work out a reasonable curriculum that includes them. One crucial thing to realize is that it will change the material being presented. For example, in the springmass problem, if one starts to look at the phasespace of the moving mass, where the computer graphics can be a real asset, then the computer is being used in a valuable way; and if one experiments with different values of the damping constant, and deduces the existence of a "critically damped" spring, then something deeper has been added to the student's understanding.
This last example is tending toward a labbased course, however, which is quite different from a lecturebased one. In my opinion, this is the setting in which the true value of computers in the classroom will become apparent. We already have several courses at Union that have moved in this direction, for example the integrated mathphysics sequence. In such courses, the students develop and discover at least some of the material of the course on their own; but this takes much more time than having the professor lecture about it. There is a tradeoff that must take place, namely, we exchange some breadth of material for depth of understanding. Such changes take considerable planning, and can not be done lightly. In the mathematics curriculum, for example, one can not simply leave out a topic in
Math 12 , say, without it having an effect onMath 17 , so coordination among all the courses is required, and that means cooperation among course instructors as well.Furthermore, making such significant course changes does not fall into the category of "standard course maintenance" but is essentially the creation of new courses. I have had the opportunity to develop a prototype labbased multivariable calculus course, so I speak from experience when I say that it is an enormously timeconsuming process. For the course I taught, a onehour lab usually required at least three solid days of preparation by each of the two instructors involved, and in the end, only about half of the labs were considered successful enough to merit continued development. This is something that simply can't be done during the term, when you are meeting with the class every other day. If we are to provide quality courses that include the use of computer labs with significant pedagogical value, we, as a college, need to come up with some creative ways of making that possible. I'm looking forward to contributing my experience to this effort.

