Engaging Students through Projects

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 they’ve 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 2004–05. 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. random Problem: How many regions are formed by 5 planes in space? Start with wild guesses: 10, 25, 32, …

15. 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: ???

16. 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

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

20. START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD Test your guess

21. START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD Test your guess

22. GUESS A FORMULA

23. GUESS A FORMULA

24. GUESS A FORMULA nk–1-dimensional hyperplanes in k-dimensional space cut it into:

25. GUESS A FORMULA nk–1-dimensional hyperplanes in k-dimensional space cut it into: Now prove it!

26. GUESS A FORMULA nk–1-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?

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 2n+1. 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.