Title
Dance Music (Jo Ann Ellerbrock)
Goals
(1) Students will explore group-ranking methods, approval voting, and weighted voting as it relates to social choice.
(2) Students will understand how elections can be manipulated.
(3) Students will explore the pros and cons of the various group-ranking methods.
Abstract
This activity, which is set in the context of selecting music for a school dance, asks students to use group-rankings to make a decision. As students proceed through the questions, they will see how the outcome of an election may very well depend on the voting procedure or agenda as much as on the voters choice.
Problem Statement
The problem of turning individual preferences for different outcomes into a single choice by the group as a whole has occupied social philosophers and political scientists for centuries. Few people realize that the voting method they use can determine the outcome of an election, and that the deciding factors are subject to manipulation. Help students understand that it is no easy matter to ascertain the "true will of the people", and that the problem of unifying individual preferences into one particular choice for the whole group is essentially unsolvable.
Instructor Suggestions
(1) Discuss with students the problems of deciding a vote with majority rule techniques when there is more than one choice. Explain how more than two candidates can make it possible that the person with the highest tally will not actually hold a majority of all votes case.
(2) Distribute the "Dance Music"; activity sheet.
(3) Have students propose some solutions (methods) to deciding what kind of music would meet the most student's approval.
(4) Discuss some of the methods of voting methods that are available. (Plurality, elimination and runoffs, sequential pairwise comparisons, various weighted or scoring schemes, approval voting, etc.)
(5) Have students form small groups to evaluate the fairness of each method. Each group should report their decision on the most appropriate type of music to the class.
Materials
Dance Music activity sheet, pencils, erasers, calculators
Time
The basic problem can be accomplished in one to two class periods depending on the amount of discussion on "fairness" and the amount of pre-knowledge the students have in group ranking methods.
Mathematics Concepts
Discrete Mathematics Concepts
Group Ranking, Plurality Winner, Majority Winner, Borda Method, Runoff Method, Sequential Runoff Method, Condorcet Method
Related Mathematics Concepts
Matrices, Permutations and Combinations, Weighted Averages
NCTM Standards Addressed
Problem Solving, Communication, Reasoning, Connections, Discrete Mathematics
Colorado Model Content Standards Addressed
Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity could be integrated into first year Algebra activities on the use of matrices or into any class unit on statistics that involves weighted averages. It also integrates well with Social Studies units involving voting and other citizenship issues.
Further Investigation
This problem lends itself well to a variety of explorations into topics such as insincere voting, bogus amendments, Arrow's Conditions, Weighted Voting.
Variations/Comments
This exercise is modified from a problem found in For All Practical Purposes: Introduction to Contemporary Mathematics, Second Edition, a project of the Consortium for Mathematics and Its Applications (COMAP), 57 Bedford Street, Suite 210, Lexington, MA 02173.
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W.H. Freeman and Company
Kenney, M.J., & Hirsch, C.R. (Eds.). (1991). Discrete mathematics across the curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
Steen, Lynn A. (Ed.). (1991). For All Practical Purposes. New York: W.H. Freeman and Company