Teaching Combinatorics

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Teaching Combinatorics in a Discrete Math Class
David M. Bressoud Macalester College St. Paul
Minnesota MathFest Albuquerque NM August 5 2005
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MATH 136 DISCRETE MATHEMATICS An introduction
to the basic techniques and methods used in
combinatorial problem-solving. Includes basic
counting principles induction logic recurrence
relations and graph theory. Every
semester. Required for a major or minor in
Mathematics and in Computer Science. I teach it
as a project-driven course in combinatorics
number theory. Taught to 74 students 3 sections
in 200405. More than 1 in 6 Macalester students
take this course.
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Let us teach guessing MAA video George Pólya
1965

• Points
• Difference between wild and educated guesses
• Importance of testing guesses
• Role of simpler problems
• Illustration of how instructive it can be to
  discover that you have made an incorrect guess

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Let us teach guessing MAA video George Pólya
1965

• Points
• Difference between wild and educated guesses
• Importance of testing guesses
• Role of simpler problems
• Illustration of how instructive it can be to
  discover that you have made an incorrect guess

• Preparation
• Some familiarity with proof by induction
• Review of binomial coefficients

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Problem How many regions are formed by 5 planes
in space
Start with wild guesses 10 25 32
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Problem How many regions are formed by 5 planes
in space
Start with wild guesses 10 25 32
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Problem How many regions are formed by 5 planes
in space
Start with wild guesses 10 25 32
Simpler problem 0 planes 1 region 1 plane 2
regions 2 planes 4 regions 3 planes 8 regions 4
planes
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Problem How many regions are formed by 5 planes
in space
Start with wild guesses 10 25 32
Simpler problem 0 planes 1 region 1 plane 2
regions 2 planes 4 regions 3 planes 8 regions 4
planes
Educated guess for 4 planes 16 regions
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TEST YOUR GUESS
Work with simpler problem regions formed by
lines on a plane
0 lines 1 region 1 line 2 regions 2 lines 4
regions 3 lines
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TEST YOUR GUESS
Work with simpler problem regions formed by
lines on a plane
0 lines 1 region 1 line 2 regions 2 lines 4
regions 3 lines
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5
1
7
2
4
3
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START WITH SIMPLEST CASE USE INDUCTIVE REASONING
TO BUILD
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START WITH SIMPLEST CASE USE INDUCTIVE REASONING
TO BUILD
Test your guess
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START WITH SIMPLEST CASE USE INDUCTIVE REASONING
TO BUILD
Test your guess
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GUESS A FORMULA
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GUESS A FORMULA
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GUESS A FORMULA
n k1-dimensional hyperplanes in k-dimensional
space cut it into
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GUESS A FORMULA
n k1-dimensional hyperplanes in k-dimensional
space cut it into
Now prove it!
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GUESS A FORMULA
n k1-dimensional hyperplanes in k-dimensional
space cut it into
Now prove it!
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This PowerPoint presentation and the Project
Description are available at www.macalester.edu/b
ressoud/talks