Teaching style for CMPE 16 (Applied Discrete Math)

Kevin Karplus
Draft version 9 Feb 1998

This document is an attempt to describe the teaching style I have adopted for teaching Computer Engineering 16 (Applied Discrete Math) and why I use it for this class.

Applied discrete math is a required course for computer engineering majors and computer and information sciences majors. It has no pre-requisites, but assumes reasonable competence at high school algebra. Previous exposure to simple proof techniques (such as would be found in a high school geometry class) should also be assumed, but I have found students woefully lacking in this aspect.

Because the class is required for all computer science and computer engineering students, class sizes are large (80--120 students) and the range of skills in the class varies widely. The material also varies considerably in difficulty, including fairly easy subjects like definitions of sets and functions, and ones that many students have great difficulty with, like mathematical induction and recursion. Indeed, from a computer science perspective, the mathematical induction and recursion are the most important concepts in the class, as they form the basis for almost all theoretical computer science.

The first two times I taught this class, I had very little advance notice, and so I adopted a teaching style that did not require extensive advance preparation. The third time (Fall 1997), although I had ample prior notice, I decided to use the same teaching style, as it seemed to work fairly well, and was attractive for pedagogic reasons, not just pragmatic ones.

The large class size forces the use of a lecture hall for presenting material, but I do not deliver traditional lectures. In the traditional scheme, the professor lectures on some material, then assigns some homework based on it, then tests the students to see if they have learned the material.

I turn this approach around---I cover in class only material that students have already attempted to work on. I assign reading and homework first, have the students attempt to work the problems based on what they read, then answer questions about the examples or exercises that the students find particularly difficult. The entire class time is directed toward the areas where students have questions, with little time wasted on going over things they understand perfectly well.

Few students in the traditional approach read the book, as they expect to have everything presented in class. Since there are not enough lecture hours to thoroughly cover all the material in most of our classes, it is essential that students learn to rely on books as a primary source early in their careers.

The lecture style is good for transferring information, but not for transferring skills, and this class is more about developing basic mathematical skills than about specific information. Students learn these skills (writing proofs, solving recurrence relations, expressing concepts in logic notation, counting permutations, and so forth) by practicing them, and by getting some guidance when they get stuck.

Practice is very different from watching someone else do similar problems, and is even more different from watching some present a slick cleaned-up presentation of the solution with no hint as to how the solution was created. In this class, I try to avoid the pitfalls of lecturing by having the students attempt the problems before I solve them, and by doing "live-action" math in front of the class. That is, I do not pre-prepare several examples that I wish to show the students---instead I attempt to solve whatever problem they are having trouble with with, showing them how I think about such problems as I work it out.

In fact, I manage to present almost the same material that a traditional lecture format would entail---the main difference is that I present the material in response to student requests, rather than as canned lectures.

Live-action math requires less preparation but a more thorough knowledge of the subject than traditional lectures. The resulting presentations are not nearly as slick as prepared lectures, but give students a better feel for how problems are really solved, and how to get out of dead-end approaches.

Live-action math does require more alertness than traditional lecturing--I'd probably be unable to teach in this style before ten in the morning, and Fall 1997 was difficult as I had colds or flu for most of the quarter and was exhausted most of the time. It can be very embarrassing to have difficulty solving a straight-forward problem, so there is strong incentive to be alert for the class. Only once have I gotten stuck so badly that I did not finish the problem by the end of the class, and in that case one of the students showed me a solution right after class---I had her present it to the class in the next lecture.

Although no formal study of student outcomes has been made, student performance on exams seems to have been about the same with this approach as with a more traditional lecture presentation. This is not to say that all the students like my style of presentation---indeed many students are indignant about having to read the book and think about the material before class or about my not giving them a simple formula that they can plug into to solve the problems. Some of the student feedback has been quite negative. There is a general student perception (incorrect in my view), that my job is to tell them precisely what they need, rather than to help them learn. The teaching style would probably be more palatable to upper-division students in a smaller class, but I have not had an opportunity to test this yet.

Generally the top students are happier with my style than the students having more difficulty with the material---perhaps because they have greater confidence about asking questions, perhaps because they get less frustrated in their initial attacks on the problems, or perhaps because they actually attempt to do the work before class, as is required for this style to work.

I alternate teaching the course with Tracy Larrabee, who uses a more traditional approach, so that students who do poorly under one teaching style have the opportunity to try again with the other style. In Winter 1998 we are team-teaching the course, alternating lectures. I think this approach will work out best for the students, as each style has students for which it works better.


Kevin Karplus
Computer Engineering
University of California, Santa Cruz
Santa Cruz, CA 95064
USA
karplus@cse.ucsc.edu
(408) 459-4250