This quarter I am again teaching introductory discrete mathematics to a group of computer science and information systems students. Here is a commentary about the students in a similar course I taught some years ago: "Very few of my students had an intuitive feel for the equivalence between a statement and its contrapositive or realized that a statement can be true and its converse false. Most students did not understand what it means for an if-then statement to be false, and many also were inconsistent about taking negations of and and or statements. Large numbers used the words "neither-nor" incorrectly, and hardly any interpreted the phrases "only if" or "necessary" and "sufficient" according to their definitions in logic. All aspects of the use of quantifiers were poorly understood, especially the negation of quantified statements and the interpretation of multiply quantified statements. Students neither were able to apply universal statements in abstract settings to draw conclusions about particular elements nor did they know what processes must be followed to establish the truth of universally (or even existentially) quantified statements. Specifically, the technique of showing that something is true in general by showing that it is true in a particular but arbitrarily chosen instance did not come naturally to most of my students. Nor did many students understand that to show the existence of an object with a certain property, one should try to find the object."[1] It is true that not all students exhibit the difficulties described in this quotation. But at my university the average SAT/ACT scores are well above national averages, and yet all but a handful of the hundreds of the discrete mathematics students I have taught over the past two decades have had the difficulties described above. Some cognitive psychologists estimate that only about 1% to 4% of the population are able, without explicit training, to correctly apply the principles of formal logic in situations where it is appropriate to do so. The problem is that students of computer science and information systems are severely hampered by weakness in their logical/analytical abilities. A significant fraction, probably a significant majority of computer science students nationwide, do not enter college with the thinking skills that would enable them to operate most effectively in a technically sophisticated environment. As I see it, the primary value of a discrete math course that is specifically addressed to freshman and sophomore students is that it can be structured so as to address students' fundamental misconceptions and difficulties with logical reasoning and improve their general analytical abilities. It is not easy to change students' deeply embedded mental habits. Consider a student who just looks blank when you come to the concluding punch line of a trivial two or three line proof by contradiction. To a person who comes by logical reasoning skills naturally, understanding such a proof may seem to require no more than common sense. But as the saying goes, common sense is not so common. The student with the blank look does not suddenly attain enlightenment by listening to three hours of lectures setting forth the rules of formal logic. To be effective, development of reasoning skills must be a theme that runs through a whole discrete math course or course sequence. And, ideally, faculty teaching subsequent mathematics and computer science courses would continue to help students improve in this area. The more computer science and information systems enrollments grow, the more schools are faced with students who have not spontaneously developed reasoning skills through exposure to standard mathematics and computer science courses. If a certain percentage of these students are led to think more logically by taking, early in their careers, a discrete mathematics course that emphasizes deductive reasoning skills, computer science education as a whole will benefit.

REFERENCE

1. The Logic of Teaching Calculus. In Toward a Lean and Lively Calculus:

Report of the Conference Workshop to Develop Curriculum and Teaching Methods

for Calculus at the College Level. Ronald G. Douglas, ed. Washington, D.C.:

Mathematical Association of America, 1987, 41-59.

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