MAA PREP Workshop

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MAA PREP Workshop Valparaiso University June 4,
2003 
Thoughts on Teaching Discrete Mathematics
Susanna S. EppDePaul Universitysepp_at_condor.depau
l.edu
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Plan of Talk

• Theme of a Discrete Math Course
• Syllabus Issues
• A Few Details

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

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• 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.

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• 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.

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• For the human soul is hospitable, and will
• entertain conflicting sentiments and
• contradictory opinions with much impartiality.
• George Eliot
• Proem to Romola
• 1862-63

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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

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

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Syllabus Issues

• DePaul Discrete Mathematics I
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• Review Guide for DM I
• DePaul Discrete Mathematics II
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• Review Guide for DM II
• Syllabi from other institutions
• Reference1
• Reference2

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A Few Details

• Discrete Math Applets on the Web
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• Unifying topics and moments
• Logic, proof, and number theory
• Euclidean algorithm and proof of correctness
• Number of edges of Kn
• 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
•