Some Thoughts on Teaching Discrete Mathematics

Nobo Komagata
Department of Computer Science
The College of New Jersey
December 30, 2003 (Draft 2)

1. Background and Motivations

It has been pointed out to me that Discrete Mathematics is a difficult course to teach especially when it is offered during the freshman year.  I was told that the materials are too abstract and dry for most students.  In addition, many students seem to perceive the list of topics very disconnected.  As a computer scientist and a person who enjoys Discrete Math, this is a rather sad situation.  We can say, "Discrete Math will be extremely useful when you learn more advanced subjects in Computer Science."  This does not seem to convince and motivate many students.  While the discussion on what to cover in a Discrete Math course and what kind of examples are more effective is an important one, the focus of this essay will be on how to teach Discrete Math, esp., how to connect the topics, hoping that more students are convinced of the usefulness of Discrete Math very early in the course.

At The College of New Jersey, there is an on-going college-wide transformation of all the programs and curricula into more learning-centered ones (cf. teaching-centered ones) [e.g., Fink 2003, Huba and Freed, 2000].  Although a lot of questions and issues have been raised, I acknowledge that I do want to run my classes better and my students perform better.  That is, there certainly are things that can be improved.  The main thing I tried in the past few years is to re-examine and align the learning goals, student assessment, and learning activities of my courses (regrettably, my older syllabi failed to integrate assessment tools that would directly check learning goals).  So, I am inclined to discuss teaching Discrete Math also in the context of the transformative change at my institution.  Note that our Discrete Math is now a sophomore course (moved from Freshman a few years ago).

Since I am new to teaching Discrete Math (since Fall 2003), there will be a lot of things I may be missing and I will need to learn.  However, I felt that I must write this essay just to clarify my thoughts and to hopefully share them with some other people.

2. My Approach: Discrete Math for Formal Modeling of the Real World

It is not my intention to insist or defend the following aspect as the core of Computer Science: "to transform real-world problems into computational ones and solve the computational problems" (such a discussion would belong elsewhere).  However, I wanted to start from some general property of Computer Science which can be used in designing a Discrete Math course (in the spirit of learning-centered approach) and also wanted to emphasize the connection between computation and the real world.

The way I perceive Discrete Math is a means to model real world, formally.  That is, I view Discrete Math as the first step of transforming real-world problems.  This places Discrete Math at a unique position in a Computer Science curriculum, in contrast to many other courses which focus on solving computational problems.  When students want to write a program to solve some real-world problem, they will need to apply Discrete Math at some point.  For example, I asked my students, "in order to write a program to automatic routing software (like the one on MapQuest or a GPS), how would you represent the necessary information? What about the game of musical chairs, social dance steps, or foreign policies?"  At the heart of such a modeling process, I see the logic-structure connection (at a more abstract level than data structures).  That is, the essence of mathematical modeling can be seen as specification of a (mathematical) structure involving sets, relations, and/or functions, through logical statements [e.g., Enderton, 2000 and Crossley et al., 1990; related ideas also in Suppes, 2002].

The idea of logic-structure connection is implicit in many areas; but it is rarely explicitly discussed (except mainly in mathematical logic).  But there seem to be many advantages in placing the logic-structure connection at the heart of representing real-world objects/phenomena, as described below.
One aspect I tend to emphasize is that (i) there are in general multiple ways to specify a structure and (ii) a collection of logical statements generally have multiple satisfying structure.  Although this could complicate the use of logic-structure connection, I believe it is important to emphasize that precise and correct representation of objects/phenomena is generally difficult (and in many cases impossible).  This seems to be useful for students to develop a critical attitude and appreciate good communication as well as insight into program specification.  If the idea of logic-structure connection is taken seriously, then it can be considered as a bonding principle behind all the topics and may justify a Discrete Math course on its own (although I am not against the just-in-time approach to Discrete Math).

As for the organization of the materials, I adopt a "spiral" approach, starting from informal discussion of logic-structure connection (with no math symbols) and gradually introducing formal notation.  Such an approach may not work well with the traditional course organization of materials divided into topics (Hein, 2003 is somewhat different in this respect).  However, the spiral approach seems to work well with the idea of logic-structure connection (e.g., exercise (Spring 2003)).  In my course, the notion of "sets" are visited multiple times at various points in the semester.  Such a non-traditional way of introducing materials has some potential drawbacks.  For instance, it is not very straightforward to use currently available textbooks.  However, there is no textbook focusing on the logic-structure connection anyway (I list Hein, 2003 as our text, or more as reference).  In addition, at the end of the semester, I received a few comments that the informal part of the course was vague.  But I do not necessarily take those comments negatively.  Those students must have appreciated formal representation.

I think that students in general can understand and appreciate what we can do with the logic-structure connection.  The overall response of the students has been positive.  Naturally, there still are many things I want to improve on.  But I feel that this is probably a good start and an interesting approach for discussion.

3. Non-Exam-Type Assessment

I feel that exams are not the way I want to assess students (cf. Huba and Freed, 2000).  I tried to integrate students' self-evaluation as part of student assessment (with calibration by the instructor) starting Fall 2003.  It was difficult for multiple reasons (I would be happy to discuss these if I am asked for the information).  Overall, though, the experience was very positive.  Not only I was able to confirm that this type of assessment is possible within the standard course schedule, but also that the approach can encourage many students to realize where they are and try to achieve more.  I will be pursuing to improve this apsect in my other courses.