MAA PREP Workshop Valparaiso University June 4, 2003 Thoughts on Teaching Discrete Mathematics Susanna S. Epp DePaul University

Plan of Talk Theme of a Discrete Math Course Syllabus Issues A Few Details

The mathematics profession as a whole has seriously underestimated the difficulty of teaching mathematics. Ramesh Gangolli MER Workshop May 31, 1991

Theme of a Discrete Math Course Message to the Math Thinking Group (January 2001) "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." Thesis: 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. However: It is not easy to change students' deeply embedded mental habits.

For the human soul is hospitable, and will entertain conflicting sentiments and contradictory opinions with much impartiality. George Eliot Proem to Romola

My Philosophy in a Nutshell Teach logical reasoning, not just logic as a subject Be conscious of the tension between covering topics and developing students' understanding Don't rush to present topics from an advanced perspective Be aware that most students today are not very good at algebra Summary: Much miscommunication occurs because of unjustified assumptions

Role of Logic in Proof article Use of Logic in Proof draft

Syllabus Issues DePaul Discrete Mathematics I Review Guide for DM I DePaul Discrete Mathematics II Review Guide for DM II Syllabi from other institutions ReferenceReference1 ReferenceReference2

A Few Details Discrete Math Applets on the Web Unifying topics and moments –Logic, proof, and number theoryLogic, proof, and number theory –Euclidean algorithm and proof of correctness –Number of edges of K n recursive approach and confirmation with induction theorem on the total degree of a graph “n choose 2” –Analyzing an algorithm counting iterations recursive thinking to obtain a recurrence relation solving a recurrence relation O-notation