Index

Discrete Mathematics Project
Election Theory Activity

Title

A Nuclear Dilemma (Dominic D. Peressini)

Goals

  1. Students will explore the concept of group-ranking, approval voting, and weighted voting as it relates to election theory.
  2. Students will be able to individually employ, and justify, a method for determining a group-ranking and apply Arrow's Conditions to their method.
  3. Students will work in small groups in order to arrive at a consensual weighted voting system and be able to explain, and justify, this system to the rest of the class.

Abstract

This activity, which is set in the context of selecting a site for storing nuclear materials (which at the time of creating this task, was actually taking place in Boulder County), asks students to determine a group-ranking based on a set of already collected preferences. Students are also asked to devise their own "fair" weighted voting system for the selection of this nuclear storage facility. This activity is envisioned to be implemented after students have been introduced to the concept of group-ranking and have explored various group-ranking methods and algorithms.

Problem Statement

Remind students that the storage of nuclear by-products and materials is a controversial issue across the world. In fact, many communities are struggling with this matter (refer to any local situations you are aware of--if near Colorado, you may want to share the newspaper articles included in the references). Let students know that today they will be examining different ways to arrive at the selection of a possible nuclear storage facility and the "fairness" of this location will be a central issue of the activity.

Instructor Suggestions

  1. Set the stage by discussing the "Problem Statement" (see above) with your students.
  2. Distribute the "A Nuclear Dilemma" activity sheet, attachment 1 and attachment 2 then allow the students to individually read and complete the first part of the activity.
  3. When all of the students are finished, have individual students share their group-ranking for the people of Boulder County who participated in the survey. Be sure to have the students identify the method they used and why.
  4. Have the students form small-groups and devise a weighted voting system that they believe is fair. and be able to explain, and justify, this system to the rest of the class.
  5. When the small-groups are finished, have a spokesperson for each and present their system and explain its fairness.
  6. Discuss the students work as it relates to rank-ordering, weighted voting, and approval-voting.

Materials

"A Nuclear Dilemma" activity sheet, attachment 1, attachment 2, chalk- or dry-erase board.

Time

Introduction of Problem Statement (5 min.), individual work (10 min.), individual presentations and discussion (10 min.), small-group work (20 min.), presentation of small-group work and large-group discussion (15 min.)

Mathematics Concepts

Discrete Mathematics Concepts

Group-Ranking, Plurality Winner, Majority Winner, Borda Method, Runoff Method, Sequential Runoff Method, Condorcet Method (paradox), Fairness, Approval Voting, Arrow's Conditions, Weighted Voting, Winning Coalitions, Power Index, Recurrence Relations

Related Mathematics Concepts:

Matrices, Permutations and Combinations

NCTM Standards Addressed:

Problem Solving, Communication, Reasoning, Connections (within mathematics and across disciplines), Algebra, Geometry, Discrete Mathematics

Colorado Model Content Standards Addressed:

Number Sense (1), Algebraic Techniques (2), Problem Solving Techniques (5), Linking Concepts and Procedures (6)

Curriculum Integration

This activity could be integrated (1) into an Algebra or Geometry class as the topic of Matrices, and their related operations, are examined--this integration would be bridged by the application of matrix modeling in order to apply Borda's method. The activity could also be integrated (2) into a traditional Advanced Algebra or Pre-Calculus class when the topics of permutations and combinations are examined--the focus would be on determining power indices by listing winning coalitions through the use of permutations/combinations.

Further Investigation

This activity can be extended in a variety of ways by using the attached demographic data to manipulate possible locations for the facility, preference schedules, and number of people responding to the survey. Ask the students to devise a set of possible, and reasonable, preference schedules.

Variations/Comments:

The first part of the activity could be used as a formal assessment. After the weighted voting systems have been presented, the students can participate in a "mock" election in order to implement their systems and experience them first-hand.
Send your variations and comments to this Web page's editor

References/Resources

Brams, S. J. (1983). Approval voting. Boston: Birkhauser.

Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W. H. Freeman and Company.

Daily Camera. (June 23, 1996). 1996 Boulder County almanac (special section). Daily Camera, pp. 1-28.

Kenney, M. J., & Hirsch, C. R. (Eds.). (1991). Discrete mathematics across the curriculum, K-12. Reston. VA: National Council of Teachers of Mathematics.

Logan, C. (1996). Flats regulators to sign cleanup plan. Weekend Colorado Daily, 104, 97, pp. 1, 5.

Lum, L., & Kurtz, D. C. (1989). Voting made easy: A mathematical theory of election procedures. Greensboro, NC: Guilford Press.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Last updated November 22, 1996