social choice in a liberal arts mathematics course

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Social Choice in a Liberal ArtsMathematics CourseGregory A. KelseyPublished online: 31 Jan 2013.

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PRIMUS, 23(2): 161–172, 2013Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970.2012.716143

Social Choice in a Liberal Arts Mathematics Course

Gregory A. Kelsey

Abstract: I present a unit on voting theory and social choice taught as the beginningof a liberal arts mathematics course for undergraduates. In this unit, students votedon the final grade calculation formula and topics to cover during the semester and thenanalyzed the results using various voting systems. I believe that by allowing my studentsto collectively decide these important aspects of the course, I empowered them andincreased their motivation. The activities also served as examples of how mathematicsplays an important role in their lives.

Keywords: Voting theory, self-determination, student motivation, quantitative literacy.

1. INTRODUCTION

Many colleges and universities offer an undergraduate course for students whoneed to fulfill a quantitative requirement. This course is often designed to pro-vide a broad survey of mathematics useful in everyday life. I call such a courseliberal arts mathematics. Others may use the terms quantitative literacy, quanti-tative reasoning, citizen mathematics, or mathematics for democracy. Whateverthey call it, mathematics departments design this course for their students whoare not expected to take mathematics again.

Popular topics for a liberal arts mathematics class include mathematicsrelated to making financial decisions and enough statistics so that students cancritically examine studies presented in the news. However, the actual contentfor a liberal arts mathematics class does not matter as much as for most othermathematics courses. Since no other course has this class as a prerequisite,teachers do not need to cover a set syllabus, and often the emphasis falls not onthe topics themselves, but rather on the critical thinking and logical reasoningskills sharpened by the mathematical work.

Address correspondence to Gregory A. Kelsey, Department of Mathematics,Trinity College, 300 Summit St., Hartford, CT 06106, USA. E-mail: [email protected]

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162 Kelsey

Teachers of liberal arts mathematics courses can encounter manychallenges not found in most other college mathematics courses. The classintentionally has a very diverse student audience. These students come frommajors which do not (by themselves) satisfy the quantitative requirements ofthe institution. Frequently, the students have had negative experiences withmathematics in the past and have chosen these majors precisely because oftheir low quantitative requirements. As a result, some students taking a liberalarts mathematics course expect the class to be dull, useless, and incomprehen-sible like their previous math experiences. Teachers face an uphill battle inmotivating their students.

In the spring semester of 2009, I taught a three-credit liberal artsmathematics course (we called it “A Mathematical World”) at the Universityof Illinois at Urbana-Champaign: a large, public, land grant university in theMidwest. The class was made up of roughly 30 students and met for 50 minutesthree times per week. My department assigned COMAP’s For All PracticalPurposes [2] as a textbook and gave me the freedom to choose which topicsfrom the text to cover in class. I chose to cover voting theory and social choicefor the first unit.

Within guidelines that I provided, I allowed the students to collectivelydecide the formula for the computation of their final grade (i.e., weigh thehomework average, quiz average, etc.) and the content of most of the semester-long course. We then analyzed these elections as a part of our unit on votingtheory. Relinquishing control over the direction of the class made a strongimpression on my students. I believe that by studying elections that affected thenature of the course, my students’ motivation and interest increased, deepeningtheir understanding of the mathematics of voting.

2. SOME EDUCATIONAL PSYCHOLOGY THEORY

To understand how to better motivate students, I turned to the educationalpsychology literature. In this section I relay some of the main points from asurvey article on student motivation by Richard M. Ryan and Edward L. Deci[4]. An interested reader will also find their seminal text on the subject to be anexcellent resource [3].

The issue of student motivation has been studied extensively in educationalpsychology. Many teachers know of the model of intrinsic versus extrinsicmotivation (known as self-determination theory in the research literature):Motivation can come from the student’s own curiosity and interest (intrinsicmotivation), or be imposed/rewarded by an external source (extrinsic motiva-tion). Well-reported studies have made many instructors aware that intrinsicmotivation leads to much stronger learning than extrinsic motivation, whichnaturally leads us to ask how to tap into these internal motivators.

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Social Choice in a Liberal Arts Mathematics Course 163

Cognitive evaluation theory (CET) asserts that activities that engenderfeelings of competency in students can enhance intrinsic motivation.Importantly, however, research shows that gains to intrinsic motivation willnot occur unless a sense of autonomy is present along with the perception ofcompetence. In other words, students experience enhanced intrinsic motiva-tion when they choose inherently interesting activities that they then performto their own satisfaction.

While CET technically does not apply to extrinsically motivated activities,the same concepts of competency and autonomy arise in promoting extrinsicmotivation with higher quality learning results. Researchers have identified anadditional factor as well: relatedness. Students who feel connected or a senseof belonging to the group imposing the activity will approach it with strongermotivation. Studies show that teachers reap the benefits of relatedness whentheir students perceive them as caring and respectful.

For teachers this research yields the following advice: to promoteautonomy, we should allow students meaningful choices about their learning.For this reason, I allowed my students to have some control over what theywould learn and how I would assess them.

3. METHODOLOGY

3.1. Choosing Course Grading

On the first day of class I had students in small groups (three or four students)discuss how to calculate the final grade of the course from homework grades,quiz grades, exam grades, and the like. I gave every student a sheet bearingthree example grading schemes (the ones labeled A, B, and C in Appendix A:Possible Grading Schemes) and asked each group to come up with their owngrading scheme subject to minimum percentages in each category (Homework:10%, Quizzes: 5%, Group Projects: 10%, Midterm Exams: 24% (8% each),Final Exam: 15%).

The groups created three new schemes (X, Y, and Z) with multiple groupsindependently designing scheme X and two designing scheme Y. Then I hadstudents vote on the grading schemes by filling out a preference list of the sixschemes and also indicating which ones they found acceptable by making a staror similar mark. A preference list (sometimes called a preference schedule) isa rank ordering of candidates from most preferred to least preferred. Usinga preference list, multiple rounds of voting (with elimination of candidatesbetween rounds) can be conducted by counting each ballot as a vote for themost preferred candidate remaining in the election. The stars are used to deter-mine the winner via approval voting, where the candidate who is acceptable tothe most voters wins.

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In the next class we started learning voting theory and over the courseof about a week covered several voting methods: plurality, plurality runoff,Condorcet, Hare (also known as instant runoff voting), Borda count, andapproval voting. As we encountered each method, I revealed the results ofour class election by that system (see Appendix B: Grading Scheme ElectionResults). In fact, student-created scheme X won under every method, so thatwas the grading scheme we used for the class.

For the next class I had the students work on our first group project, the firstfour questions of which dealt with the grade percentage election and appearin Appendix C: Grade Percentage Election Results Questions. I also directedthe students to use the schemes’ content and full election results (i.e., theinformation found in Appendices A and B) to analyze the results and votingmethods.

3.2. Choosing Course Content

In the few day lull between finishing the content to appear on the unit examfor voting theory and social choice and the day of the exam itself, I used oneclass period to do a group “jigsaw” activity to choose the topics for the restof the semester. This method forces all students to participate during the laststage of the activity [1]. I have included the instructions as Appendix D: JigsawInstructions.

I designed this jigsaw activity to have six students in each “home group.”Six units remained in our textbook, and so each group assigned one studentto each unit to learn about it. Then all the students in the class learning aboutthe same unit gathered to work together. To help them summarize their unit,I assigned guiding questions for them to answer; I have included these ques-tions as Appendix E: Unit Summary Questions. After they had summarizedtheir unit, the students reported back to their home group about their particularunit, and the whole group obtained summaries of all the units in the textbook.The students then used that information to make (hopefully) intelligent andinformed choices about what units they would like to cover next. They filledout preference lists of the units, marked for approval voting as well, and turnedthose ballots in as they left class.

I then determined the results of the vote using the various voting methodswe had covered (see Appendix F: Unit Election Results). On the exam, Iincluded these results (using randomly assigned letters to represent the unitsso that students’ preferences would not affect their analysis) and asked: “Thefollowing page has the results of our election for the next topic to be con-sidered in this course. Which unit do you think is the “will of the class” tostudy next? Why?” I graded this question based on their argument in defenseof their position, not on their opinion of which unit represented the “will of theclass.”

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Since the results of this election were more mixed than those of the gradingelection, students made many different arguments and held different opinionsfor the unit that “should” win. However, a majority of the class argued that unitD: On Size and Growth, was the “will of the class,” so we started that unit inthe next class period. Over the course of the remainder of the semester, we alsocovered units C: The Digital Revolution and F: Your Money and Resources,which won the approval voting and Borda count methods, respectively. We didnot cover any topics from the other units.

4. RESULTS

First, a disclaimer: I give here my impressions as instructor, not data collectedin a rigorous fashion. I have not determined the influence of this unit through acarefully designed, institutional review board (IRB)-approved study, but ratherby comparing my experiences in this course with my other experiences as aninstructor (I also taught calculus and geometry while a graduate student atUIUC, and I have taught liberal arts mathematics, college algebra, calculus,real analysis, and geometry as a professor at Immaculata University). I do notclaim any general inferences; I seek only to share my thoughts so that hopefullyother teachers can gain from them.

I found that starting the course by having students give input on the gradingdefied their expectations of a mathematics class. I believe that by distancingthis course from their previous math classes in this way, my students did notallow past prejudices to influence their behavior as much as they might haveotherwise. Specifically, I had an easier time convincing my students that thecourse was designed for their benefit, and they seemed more willing to makeattempts at new or different things.

This unit really helped me “sell” the core conceit of the liberal artsmathematics course to my students: namely, that mathematics can play a usefulrole in their lives after college. I value getting students on board with this ideamore than them learning any particular content of the course. By relinquishingcontrol over the course, I signaled to my students that I value their opinions.It takes only a short step to move from that point to the notion that I teachmaterial for their benefit.

I also found increased student interest when we began the elected topics.Students had previewed the topic, and so might recall one or two interestingthings about it (or at least recall that they had in the past seen one or twointeresting things about it). Also, I believe that some students felt a sense of“ownership” for one of the topics: one that they had summarized for their groupor voted for in the election. Starting “their” unit excited these students.

In terms of the voting theory material itself, my students analyzed electionsnot only in abstract examples in a textbook, but also ones in which they par-ticipated and had a vested interest in the outcome. This led to more detailed

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analysis than I saw in response to other voting theory problems. This activ-ity also helped to strike home the point that making collective decisions affectstheir lives and that math plays an important role in those social choices. My stu-dents certainly saw how it played a role in determining their grade for thecourse.

Finally, the voting theory unit really made an impression on my students,even more than the elected units. At the end of the semester, I conducted aninformal survey (i.e., one for my personal benefit and not IRB-approved forresearch). Since this was an informal survey, I will not report exact responseshere, but when I asked “Has this course changed the way that you think aboutanything?” the voting theory material stood out as the only specific topic tocome up from more than a few students. I should point out that the votingtheory unit was not popular, as only a few students mentioned it in response to“What was your favorite topic that we covered?”

Sharing the choice of content for the course did have some negative con-sequences. Most importantly, as a direct result of the topic election we did notcover any statistics. In my opinion, all college-educated students should havea basic level of statistical literacy, so I regret this omission from my course.In the future, I plan to follow the social choice unit with a unit on statistics andthen begin the elected topics. This change still allows students to control halfof the course content and also gives me some more time to prepare the electedmaterial.

ACKNOWLEDGEMENTS

The author thanks Rochelle Gutiérrez’s Fall 2008 Equity in Mathematics andScience Education and Michael Loui’s Spring 2009 College Teaching classesat UIUC for their helpful comments and suggestions. The author also givesvery warm thanks to his own Spring 2009 A Mathematical World class fortheir participation, hard work, and enthusiasm. Additionally, the author thanksthe anonymous referees for their careful consideration of earlier drafts of thisarticle.

APPENDIX A: POSSIBLE GRADING SCHEMES

Scheme A

Homework: 15%Quizzes: 10%Group Projects: 15%Midterm Exams: 36% (12% each)Final Exam: 24%

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Scheme B


Scheme C


Scheme X


Scheme Y


Scheme Z


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APPENDIX B: GRADING SCHEME ELECTION RESULTS

Plurality vote

X: 48%, Y: 21%, B: 17%, Z: 14%, A: 0%, C: 0%.

Condorcet method

X def. Y: 66% − 34%, X def. B: 79% − 21%, X def. Z: 83% − 17%, X def.C: 90% − 10%, X def. A: 93% − 7%.

Approval voting

X: 90%, Y: 79%, B: 72%, A: 30%, Z: 28%, C: 7%.

Borda scores

X: 119, Y: 110, B: 82, Z: 57, A: 49, C: 15.

Sequential voting

X wins under any agenda by the results found above for the Condorcet Method.

Hare system

Round 1: X: 48%, Y: 21%, B: 17%, Z: 14%, A: 0%, C: 0%; A and C eliminated.Round 2: X: 48%, Y: 21%, B: 17%, Z: 14%; Z eliminated.Round 3: X: 48%, Y: 31%, B: 21%; B eliminated.Round 4: X: 66%, Y: 34%; Y eliminated.Round 5: X: 100%; X wins.

Plurality runoff

Round 1: X: 48%, Y: 21%, B: 17%, Z: 14%, A: 0%, C: 0%.Round 2: X: 66%, Y: 34%.

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APPENDIX C: GRADE PERCENTAGE ELECTION RESULTSQUESTIONS

1. It seems clear that scheme X is the favorite scheme of the class. By howmuch more do you think the class favored it than the other schemes, a greatdeal or only a little? Cite specific evidence that backs up your opinion.

2. We noted in class that the schemes X, Y, and B all seemed to do the best,and we theorized that this was due to the fact that they share the 15% FinalExam. How do you think the class felt about the 24% Final Exam and 30%Final Exam? How important do you see the Final Exam Percentage beingin people’s votes? Use specific results that demonstrate your answer.

3. Suppose that the math department ruled that 15% was too low for a finalexam. Given our election, what grading scheme would you recommend?What evidence supports your view? What evidence detracts from yourargument, and why does it not change your opinion?

4. Which of these voting methods gives results that you feel best represent “thewill of the class”? What about the data do you find so convincing? Do youthink that this voting method is “the best” or that it just happened to workout well in our particular setting? Why?

APPENDIX D: JIGSAW INSTRUCTIONS

This activity is meant to introduce you to the many topics covered in ourtextbook. Each member of your group will learn about one unit from the bookand then you will all report back so that everyone has a general sense of thetopics. You will receive a participation grade for your work in this activity.

Step 1: Divide up the topics

There are six units remaining in our textbook:

1. Management Science – efficiency, scheduling, and the like.2. Statistics and Probability – obtaining, analyzing, and representing data.3. Fairness and Game Theory – a natural successor to the voting theory unit.4. The Digital Revolution – coding theory and identification numbers.5. Size and Growth – geometry and art.6. Money and Resources – economics of finance and resources.

Your group must assign at least one student to each of these units. If yourgroup cannot come to a quick agreement, the group member with the birthdaythat falls earliest in the year makes the final call.

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Step 2: Investigate the topics

Each of you will join the others studying the same unit and will work togetherto fill out the relevant page of this packet. Feel free to make additional notesto help your groupmates understand what the unit is about. You will have20–25 minutes for this portion of the activity.

Step 3: Share your findings

After you have completed your study of your chapters, your group willreconvene to swap information. Help your groupmates to fill out their packetand share any additional information you feel is relevant about the unit you con-sidered. Also, fill out the pages on the units you did not cover by listening toyour fellow group members describe their work. You will have 15–20 minutesfor this portion of the activity.

Step 4: Vote

Now that you have some knowledge on all of the topics covered by thetextbook, vote for which ones you want to cover next in the course. Below,write down your preference list of the units and mark your approval with an“X” or a star. You will have about 5 minutes for this final portion of the activity.

APPENDIX E: UNIT SUMMARY QUESTIONS

1. What are the “big questions” that this unit seeks to answer?2. What skills would you learn in this unit?3. In what jobs/fields could the mathematics of this unit be applied?4. Find one problem considered by this unit that strikes you as particularly

interesting and describe it.

APPENDIX F: UNIT ELECTION RESULTS

Key: (not included on the exam)

A: Fairness and Game TheoryB: Management ScienceC: The Digital RevolutionD: On Size and Growth

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E: Statistics: The Science of DataF: Your Money and Resources

Plurality vote

D: 35%, F: 30%, A: 13%, C: 9%, B: 9%, E: 4%.

Condorcet method

D def. C: 52% − 48%, D def. F: 57% − 43%, D def. B: 65% − 35%, D def.E: 65% − 35%, D def. A: 78% − 22%.

Approval voting

C: 83%, F: 74%, D: 70%, B: 61%, A: 34%, E: 30%.

Borda scores

F: 73, C: 69, D: 68, B: 47, E: 38, A: 35.

Sequential pairwise voting

D wins under any agenda by the results found above for the Condorcet Method.

Hare system

Round 1: D: 35%, F: 30%, A: 13%, C: 9%, B: 9%, E: 4%; E eliminated.Round 2: D: 39%, F: 30%, A: 13%, C: 9%, B: 9%; B and C eliminated.Round 3: D: 52%, F: 35%, A: 13%; A eliminated.Round 4: D: 57%, F: 43%; F eliminated.Round 5: D: 100%; D wins.

Plurality runoff

Round 1: D: 35%, F: 30%, A: 13%, C: 9%, B: 9%, E: 4%.Round 2: D: 57%, F: 43%.

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REFERENCES

1. Aronson, E., N. Blaney, C. Stephan, J. Sikes, and M. Snapp. 1978. TheJigsaw Classroom. Beverly Hills, CA: Sage.

2. Consortium for Mathematics and its Applications. 2003. For All PracticalPurposes: Mathematical Literacy in Today’s World, Sixth Edition. NewYork, NY: W. H. Freeman and Company.

3. Deci, E. L. and R. M. Ryan. 1985. Intrinsic Motivation and Self-Determination in Human Behavior. New York, NY: Plenum.

4. Ryan, R. M. and E. L. Deci. 2000. Intrinsic and extrinsic motiva-tions: Classic definitions and new directions. Contemporary EducationalPsychology. 25: 54–67.

BIOGRAPHICAL SKETCH

Gregory A. Kelsey is a Visiting Assistant Professor in the Department ofMathematics at Trinity College in Hartford, CT. As a mathematics researcher,he is interested in problems related to self-similar groups, especially theirapplications to complex dynamics. As a mathematics teacher, he is interestedin problems related to higher-order thinking skills, knowledge retention, andmotivation.

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social choice in a liberal arts mathematics course

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