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, …
random
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?
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: ???
random
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
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
6
1 line: 2 regions
1
2 lines: 4 regions
3 lines: ???
2
5
7
3
4
START WITH SIMPLEST CASE
USE INDUCTIVE REASONING TO BUILD
n
Space cut by n Plane cut by n
planes
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 Plane cut by n
planes
lines
Line cut by n
points
0
1
1
1
1
2
2
2
2
4
4
3
3
8
7
4
11
5
4
5
Test your guess
6
START WITH SIMPLEST CASE
USE INDUCTIVE REASONING TO BUILD
n
Space cut by n Plane cut by n
planes
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
Test your guess
6
GUESS A FORMULA
n
0
1
2
3
4
5
6
points
on a line
lines on
a plane
planes
in space
1
2
3
4
5
6
7
1
2
4
7
11
16
22
1
2
4
8
15
26
42
 n
 k 
GUESS A FORMULA
n
0
1
2
3
4
5
6
points
on a line
lines on
a plane
planes
in space
1
2
3
4
5
6
7
1
2
4
7
11
16
22
1
2
4
8
15
26
42
k
n
0
1
2
3
4
5
6
0
1
1
1
1
1
1
1
1
0
1
2
3
4
5
6
2
0
0
1
3
6
3
0
0
0
1
4
10 10
4
0
0
0
0
1
5
15 20 15
5
0
0
0
0
0
1
6
6
0
0
0
0
0
0
1
GUESS A FORMULA
n k–1-dimensional hyperplanes in k-dimensional space
cut it into:
 n  n  n
 n
 0    1    2   L   k  regions.
GUESS A FORMULA
n k–1-dimensional hyperplanes in k-dimensional space
cut it into:
 n  n  n
 n
 0    1    2   L   k  regions.
Now prove it!
GUESS A FORMULA
n k–1-dimensional hyperplanes in k-dimensional space
cut it into:
 n  n  n
 n
 0    1    2   L   k  regions.
Now prove it!
Show that if R n, k   # of regions with n hyperplanes
in k-dimensional space, then
R(n, k)  R(n  1, k)  R(n  1, k  1).
What do you have to assume about k  1-hyperplanes in
k-dimensional space?
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 4
5 6 7 8 9
10 11 12 13 14
15 16 17 18 19
20 21 22 23 24
25 26 27 28 29
30 31 32 33 34
X
0
1 2
3 4
5 6 7 8 9
X
X
10
11 12 13 14
X
15
16 17 18 19
20
21 22 23 24
X
X
25
26 27 28 29
X
30
31 32 33 34
X
0
1 2
3 4
5 6 7 X8 9
X
X
X
10
11 12 13
14
X
X
15
16 17 18
19
20
21 22 23
24
X
X
X
X
25
26 27 28
29
X
X
30
31 32 33
34
X
0
1 2
3 4
5 6 7 X8 9
X
X
X
10
11 12 13
14
X
X
X
15
16
17 18
19
20
21
22 23
24
X
X
X
X
X
X
X
X
25
26
27 28
29
X
X
X
X
X
30
31
32
33
34
X
0
1 2
3 4
5 6 7 X8 9
X
X
X
10
11 12 13
14
X
X
X
15
16
17 18
19
20
21
22 23
24
X
X
X
X
X
X
X
X
25
26
27 28
29
X
X
X
X
X
30
31
32
33
34
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.