David M. BressoudMacalester College, St. Paul, MNProject 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% eachIf 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.
MATH 136 DISCRETE MATHEMATICSAn 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.
Let us teach guessing MAA video, George Plya, 1965Points:Difference between wild and educated guessesImportance of testing guessesRole of simpler problemsIllustration of how instructive it can be to discover that you have made an incorrect guess
Let us teach guessing MAA video, George Plya, 1965Points:Difference between wild and educated guessesImportance of testing guessesRole of simpler problemsIllustration of how instructive it can be to discover that you have made an incorrect guessPreparation:Some familiarity with proof by inductionReview 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,
Simpler problem:0 planes: 1 region1 plane: 2 regions2 planes: 4 regions3 planes: 8 regions4 planes: ???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?Simpler problem:0 planes: 1 region1 plane: 2 regions2 planes: 4 regions3 planes: 8 regions4 planes: ???Start with wild guesses: 10, 25, 32, Educated guess for 4 planes: 16 regions
TEST YOUR GUESSWork with simpler problem: regions formed by lines on a plane:0 lines: 1 region1 line: 2 regions2 lines: 4 regions3 lines: ???
TEST YOUR GUESSWork with simpler problem: regions formed by lines on a plane:0 lines: 1 region1 line: 2 regions2 lines: 4 regions3 lines: ???1234567
START WITH SIMPLEST CASEUSE INDUCTIVE REASONING TO BUILD
nSpace cut by n planesPlane cut by n linesLine cut by n points01111222244338744556
START WITH SIMPLEST CASEUSE INDUCTIVE REASONING TO BUILDTest your guess
nSpace cut by n planesPlane cut by n linesLine cut by n points0111122224433874411556
START WITH SIMPLEST CASEUSE INDUCTIVE REASONING TO BUILDTest your guess
nSpace cut by n planesPlane cut by n linesLine cut by n points011112222443387441511556
GUESS A FORMULA
npoints on a linelines on a planeplanes in space0111122223443478451115561626672242
GUESS A FORMULA
npoints on a linelines on a planeplanes in space0111122223443478451115561626672242
01234560100000011100000212100003133100041464100515101051061615201561
GUESS A FORMULAn k1-dimensional hyperplanes in k-dimensional space cut it into:
GUESS A FORMULANow prove it!n k1-dimensional hyperplanes in k-dimensional space cut it into:
GUESS A FORMULANow prove it!n k1-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?
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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-shufflesIn-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.
