engaging students through projects

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David M. Bressoud Macalester College, St. Paul, MN Project NExT-WI, October 6, 2006

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Engaging Students through Projects. 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. - PowerPoint PPT Presentation

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Page 1: Engaging Students through Projects

David M. Bressoud

Macalester College, St. Paul, MN

Project NExT-WI, October 6, 2006

Page 2: Engaging Students through Projects

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

Page 3: Engaging Students through Projects

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

•Try to avoid doing everything new in any course.

Page 4: Engaging Students through Projects

•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.

Page 5: Engaging Students through Projects

•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

Page 6: Engaging Students through Projects

•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

Page 7: Engaging Students through Projects

•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

Page 8: Engaging Students through Projects

•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

Page 9: Engaging Students through Projects

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.

Page 10: Engaging Students through Projects

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.

Page 11: Engaging Students through Projects

“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

Page 12: Engaging Students through Projects

“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

Page 13: Engaging Students through Projects

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

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

Page 14: Engaging Students through Projects

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

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

random

Page 15: Engaging Students through Projects

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

Page 16: Engaging Students through Projects

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

Page 17: Engaging Students through Projects

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: ???

Page 18: Engaging Students through Projects

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

Page 19: Engaging Students through Projects

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

Page 20: Engaging Students through Projects

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

Page 21: Engaging Students through Projects

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

Page 22: Engaging Students through Projects

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

Page 23: Engaging Students through Projects

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

⎛⎝⎜

⎞⎠⎟

Page 24: Engaging Students through Projects

GUESS A FORMULA

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

n0

⎛⎝⎜

⎞⎠⎟ +

n1

⎛⎝⎜

⎞⎠⎟ +

n2

⎛⎝⎜

⎞⎠⎟ +L +

nk

⎛⎝⎜

⎞⎠⎟ regions.

Page 25: Engaging Students through Projects

GUESS A FORMULA

n0

⎛⎝⎜

⎞⎠⎟ +

n1

⎛⎝⎜

⎞⎠⎟ +

n2

⎛⎝⎜

⎞⎠⎟ +L +

nk

⎛⎝⎜

⎞⎠⎟ regions.

Now prove it!

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

Page 26: Engaging Students through Projects

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:

Page 27: Engaging Students through Projects

Stamp Problem:

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

Page 28: Engaging Students through Projects

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Page 29: Engaging Students through Projects

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Page 30: Engaging Students through Projects

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Page 32: Engaging Students through Projects

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Page 33: Engaging Students through Projects

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¢?

Page 34: Engaging Students through Projects

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.

Page 35: Engaging Students through Projects

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.