David M. Bressoud

Macalester College, St. Paul, MN

Project NExT-WI, October 6, 2006

•Do something that is new to you in every course.

•Do something that is new to you in every course.

•Try to avoid doing everything new in any course.

•Do something that is new to you in every course.

•Try to avoid doing everything new in any course.

•What you grade is what counts for your students.

•Do something that is new to you in every course.

•Try to avoid doing everything new in any course.

•What you grade is what counts for your students.

Reading mathematics, working through complex problems, communicating mathematics, using terminology correctly, constructing proofs, going back over material that has not been understood

•Do something that is new to you in every course.

•Try to avoid doing everything new in any course.

•What you grade is what counts for your students.

Reading mathematics, working through complex problems, communicating mathematics, using terminology correctly, constructing proofs, going back over material that has not been understood

•Do something that is new to you in every course.

•Try to avoid doing everything new in any course.

•What you grade is what counts for your students.

Reading mathematics, working through complex problems, communicating mathematics, using terminology correctly, constructing proofs, going back over material that has not been understood

•Do something that is new to you in every course.

•Try to avoid doing everything new in any course.

•What you grade is what counts for your students.

Reading mathematics, working through complex problems, communicating mathematics, using terminology correctly, constructing proofs, going back over material that has not been understood

What you grade is what counts for your students.

• Homework 20%

• Reading Reactions 5%

• 3 Projects 10% each

• 2 mid-terms + final, 15% each

If you hold students to high standards and give them ample opportunity to show what they’ve learned, then you can safely ignore cries about grade inflation.

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 2004–05. More than 1 in 6 Macalester students take this course.

“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

“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

Problem: How many regions are formed by 5 planes in space?

Start with wild guesses: 10, 25, 32, …

Problem: How many regions are formed by 5 planes in space?

Start with wild guesses: 10, 25, 32, …

random

Simpler problem:

0 planes: 1 region

1 plane: 2 regions

2 planes: 4 regions

3 planes: 8 regions

4 planes: ???

Problem: How many regions are formed by 5 planes in space?

Start with wild guesses: 10, 25, 32, …

random

Problem: How many regions are formed by 5 planes in space?

Simpler problem:

0 planes: 1 region

1 plane: 2 regions

2 planes: 4 regions

3 planes: 8 regions

4 planes: ???

Start with wild guesses: 10, 25, 32, …

Educated guess for 4 planes: 16 regions

random

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: ???

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: ???

1

23

4

5

6

7

START WITH SIMPLEST CASE

USE INDUCTIVE REASONING TO BUILD

n Space cut by n planes

Plane cut by n lines

Line cut by n points

0 1 1 1

1 2 2 2

2 4 4 3

3 8 7 4

4 5

5 6

START WITH SIMPLEST CASE

USE INDUCTIVE REASONING TO BUILD

n Space cut by n planes

Plane cut by n lines

Line cut by n points

0 1 1 1

1 2 2 2

2 4 4 3

3 8 7 4

4 11 5

5 6Test your guess

START WITH SIMPLEST CASE

USE INDUCTIVE REASONING TO BUILD

n Space cut by n planes

Plane cut by n lines

Line cut by n points

0 1 1 1

1 2 2 2

2 4 4 3

3 8 7 4

4 15 11 5

5 6Test your guess

GUESS A FORMULA

n points on a line

lines on a plane

planes in space

0 1 1 11 2 2 22 3 4 43 4 7 84 5 11 155 6 16 266 7 22 42

GUESS A FORMULA

n points on a line

lines on a plane

planes in space

0 1 1 11 2 2 22 3 4 43 4 7 84 5 11 155 6 16 266 7 22 42

0 1 2 3 4 5 60 1 0 0 0 0 0 01 1 1 0 0 0 0 02 1 2 1 0 0 0 03 1 3 3 1 0 0 04 1 4 6 4 1 0 05 1 5 10 10 5 1 06 1 6 15 20 15 6 1

kn

nk

⎛⎝⎜

⎞⎠⎟

GUESS A FORMULA

n k–1-dimensional hyperplanes in k-dimensional space cut it into:

n0

⎛⎝⎜

⎞⎠⎟ +

n1

⎛⎝⎜

⎞⎠⎟ +

n2

⎛⎝⎜

⎞⎠⎟ +L +

nk

⎛⎝⎜

⎞⎠⎟ regions.

GUESS A FORMULA

n0

⎛⎝⎜

⎞⎠⎟ +

n1

⎛⎝⎜

⎞⎠⎟ +

n2

⎛⎝⎜

⎞⎠⎟ +L +

nk

⎛⎝⎜

⎞⎠⎟ regions.

Now prove it!

n k–1-dimensional hyperplanes in k-dimensional space cut it into:

GUESS A FORMULA

n0

⎛⎝⎜

⎞⎠⎟ +

n1

⎛⎝⎜

⎞⎠⎟ +

n2

⎛⎝⎜

⎞⎠⎟ +L +

nk

⎛⎝⎜

⎞⎠⎟ regions.

Now prove it!Show that if R n,k( )=# of regions with n hyperplanesin k-dim ensional space, then

R(n,k)=R(n−1,k)+ R(n−1,k−1).What do you have to assum e about k−1-hyperplanes in k-dim ensional space?

n k–1-dimensional hyperplanes in k-dimensional space cut it into:

Stamp Problem:

What is the largest postage amount that cannot be made using an unlimited supply of 5¢ stamps and 8¢ stamps?

0 1 2 3 45 6 7 8 9

10 11 12 13 1415 16 17 18 1920 21 22 23 2425 26 27 28 2930 31 32 33 34M M M M M

0 1 2 3 45 6 7 8 9

10 11 12 13 1415 16 17 18 1920 21 22 23 2425 26 27 28 2930 31 32 33 34M M M M M

X

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0 1 2 3 45 6 7 8 9

10 11 12 13 1415 16 17 18 1920 21 22 23 2425 26 27 28 2930 31 32 33 34M M M M M

X

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0 1 2 3 45 6 7 8 9

10 11 12 13 1415 16 17 18 1920 21 22 23 2425 26 27 28 2930 31 32 33 34M M M M M

X

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0 1 2 3 45 6 7 8 9

10 11 12 13 1415 16 17 18 1920 21 22 23 2425 26 27 28 2930 31 32 33 34M M M M M

X

X

X

X

X

X

X

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Stamp Problem:

What is the largest postage amount that cannot be made using an unlimited supply of 5¢ stamps and 8¢ stamps?

4¢ and 9¢? 4¢ and 6¢?

a¢ and b¢?

How many perfect shuffles are needed to return a deck to its original order?

In-shuffles versus out-shuffles

In-shuffles in a deck of 2n cards is the order of 2 modulo 2n+1. Out-shuffles is the order of 2 modulo 2n-1.

Tips on group work:•I assign who is in each group, and I mix up the membership of the groups.

•No more than 4 to a group, then split into writing teams of 2 each. Have at least one project in which each person submits their own report.

•Each team decides how to split up the grade.