Engaging Students

1
Engaging Students through Projects
David M. Bressoud Macalester College, St. Paul,
MN Project NExT-WI, October 6, 2006
2

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

3

• Do something that is new to you in every course.
• Try to avoid doing everything new in any course.

4

• 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.

5

• 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

6

• 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

7

• 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

8

• 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

9

• 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 theyve
  learned, then you can safely ignore cries about
  grade inflation.

10
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.
11
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

12
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

13
Problem How many regions are formed by 5 planes
in space?
Start with wild guesses 10, 25, 32,
14
Problem How many regions are formed by 5 planes
in space?
Start with wild guesses 10, 25, 32,
15
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 ???
16
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
17
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 ???
18
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 ???
6
5
1
7
2
4
3
19
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
20
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 6
Test your guess
21
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 6
Test your guess
22
GUESS A FORMULA
n points on a line lines on a plane planes in space
0 1 1 1
1 2 2 2
2 3 4 4
3 4 7 8
4 5 11 15
5 6 16 26
6 7 22 42
23
GUESS A FORMULA
n points on a line lines on a plane planes in space
0 1 1 1
1 2 2 2
2 3 4 4
3 4 7 8
4 5 11 15
5 6 16 26
6 7 22 42
0 1 2 3 4 5 6
0 1 0 0 0 0 0 0
1 1 1 0 0 0 0 0
2 1 2 1 0 0 0 0
3 1 3 3 1 0 0 0
4 1 4 6 4 1 0 0
5 1 5 10 10 5 1 0
6 1 6 15 20 15 6 1
24
GUESS A FORMULA
n k1-dimensional hyperplanes in k-dimensional
space cut it into
25
GUESS A FORMULA
n k1-dimensional hyperplanes in k-dimensional
space cut it into
Now prove it!
26
GUESS A FORMULA
n k1-dimensional hyperplanes in k-dimensional
space cut it into
Now prove it!
27
Stamp Problem What is the largest postage
amount that cannot be made using an unlimited
supply of 5 stamps and 8 stamps?
28
(No Transcript)
29
X
X
X
X
X
X
X
30
X
X
X
X
X
X
X
X
X
X
X
X
X
31
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
32
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
33
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?
34
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 2n1. Out-shuffles is the
order of 2 modulo 2n-1.
35

• 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.