interdisciplinary quantitative reasoning · 2016-11-21 · interdisciplinary quantitative reasoning...

I n n o v a t i o n a n d C u r r i c u l u m D e v e l o p m e n t a t H o l l i n s U n i v e r s i t y

I N T E R D I S C I P L I N A R Y Q U A N T I T A T I V E R E A S O N I N G

I N T E R D I S C I P L I N A R Y

Q U A N T I T A T I V E R E A S O N I N G

C A R E N L . D I E F E N D E R F E R Associate Professor of Mathematics

R U T H A L D E N D O A N Professor of History

C H R I S T I N A A . S A L O W E Y Associate Professor of Classical Studies

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An intelligent citizen reads a

newspaper account of an outbreak of disease in

a small community. How can she tell if the

number of afflicted looms out of proportion to

the expected incidence of disease? A parent

must choose whether or not his child will receive

a smallpox vaccine. How can he evaluate the

benefits and the risks of such an inoculation? An

employer asks an employee to develop a profile

of the local population to provide a foundation

for a marketing campaign. How can the employee

assess the significance of distributions of age,

race, gender, or other categories in the population?


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I N T R O D U C T I O N

In order to become effective citizens,workers, parents, advocates—indeed inorder to perform a great variety ofroles—students must become competentin using and reading quantitative data,in understanding quantitative evidence,and in applying basic quantitative andmathematical skills so that they cansolve real-life problems. In fact, as LynnSteen, past president of the MathematicalAssociation of America (MAA), notes,

Quantitatively literate citizensneed to know more than formulasand equations. They need a predisposition to look at the worldthrough mathematical eyes, to see the benefits (and risks) of thinking quantitatively aboutcommonplace issues, and toapproach complex problemswith confidence in the value ofcareful reasoning. Quantitativeliteracy empowers people by giving them tools to think forthemselves, to ask intelligentquestions of experts, and to confront authority confidently.These are skills required tothrive in the modern world.1

Quantitative understanding, also called numeracy, is traditionally taughtthrough work in mathematics and statistics courses, but it can often belearned more effectively through workin courses within the student’s ownmajor or minor discipline or in thecontext of courses of interest to the student. “To be useful for the student,numeracy needs to be learned andused in multiple contexts—in historyand geography, in economics and biology, in agriculture and culinary

arts. Numeracy is not just one amongmany subjects but an integral part of all subjects.” 2

A program that involves quantita-tive reasoning across the disciplinesgives students an opportunity to learnthe broad significance and applicabilityof quantitative reasoning and mathe-matical skills in the particular subjectsthat are meaningful, important, andinteresting to them. In this sense, quan-titative reasoning across the curriculumbecomes a partner with writing acrossthe curriculum. College teachers havetoo long heard the complaint, “This is not an English course; why shouldgrammar and style count?” The writingacross the curriculum movement hassought to address that misunderstandingon the part of students by demonstratingthat writing is a skill necessary acrossdisciplines and across jobs, responsibili-ties, and social roles. Similarly, toomany students believe that quantitativereasoning comes into play only inmathematics courses, and that onceone fulfills a math requirement, thosepesky numbers will disappear. Scholarsknow better; quantitative reasoningenters into work in fields as diverse asEnglish and biology, history andphysics, theatre and statistics.

If those who teach know that students need to become competent inquantitative reasoning, students mustgain the motivation to achieve thatcompetence. Offering a student theopportunity to analyze and apply quantitative techniques in the processof studying a subject that the studentcares about or expresses an interest in sparks such motivation. The studenthas the opportunity to see that herunderstanding of a question that engagesher will be deepened by applying quantitative techniques or perhaps that


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her ability to persuade others to accepther point of view will be strengthenedby the use of quantitative evidencecarefully presented.

H O L L I N S ’ G E N E R A L E D U C AT I O N P R O G R A M

In the fall of 1998, Hollins began work on a major overhaul of its generaleducation requirements. Under theleadership of the vice president for aca-demic affairs and the Academic AffairsCouncil, the central curricular body

of the university, the Hollins communi-ty worked together to develop a newframework to define the intellectualperspectives students should engageduring their course of study at Hollins,as well as various skills students shouldacquire. In particular, our new generaleducation requirements, Educationthrough Skills and Perspectives (ESP),allow for foundational work in the skillsareas (writing, oral communication,quantitative reasoning, and informationtechnology) as needed, and then en-courage the melding of skills acquisitionwithin courses designed to introducestudents to various perspectives (formore details see www.hollins.edu/academics/esp/esp.htm). ESP requiresthat students complete four credits in each of seven perspectives: AestheticAnalysis, Creative Expression, Ancientand/or Medieval Worlds, Modernand/or Contemporary Worlds, Scientific

Inquiry, Social and Cultural Diversities,and Global Systems. In addition, thereis a language requirement. Hollins’ faculty approved the ESP program inspring 2001, and all students enteringHollins since fall 2001 are followingthis program.

T H E Q R R E Q U I R E M E N T S

Quantitative reasoning is one of theskill areas in our new general educationprogram. We have two requirements in this area.

1. The QR Basic Skills Requirement (q)The QR Basic Skills Requirement is designed to help students gain anunderstanding of fundamental mathe-matical skills they need to be successfulin courses that require quantitative reasoning. The basic skills requirementcan be satisfied by achieving a satisfac-tory score on the Quantitative ReasoningAssessment (given to new students each fall) or by successful completionof Mathematics 100 – An Introductionto Quantitative Reasoning. A studentwho has satisfied the QR basic skillsrequirement will demonstrate a baselineunderstanding of such topics as algebra,graphing, geometry, data analysis, and linearity.

Goals of the QR Basic Skills Requirement• To understand mathematical and

statistical reasoning.• To use appropriate mathematical and/

Quantitative reasoning enters into work in fields as diverse as English and biology,

history and physics, theatre and statistics.


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or statistical tools in summarizing data, making predictions, and estab-lishing cause-and-effect relationships.

In addition to covering what maybe considered to be traditional topicsin the mathematics curriculum, ourintroductory QR course has a computerlab every week. Students learn how touse Excel in a variety of ways—to graphdata, to analyze data, to explore andmodel growth, and to learn about loansand financial planning.

2. The QR Applied SkillsRequirement (Q)The QR applied skills requirement isdesigned to provide students with the opportunity to apply quantitativeskills as they solve problems in fields of study in which they show an interest.Students can satisfy the applied skillsrequirement by passing a course desig-nated as a QR applied course.

Goals of the QR Applied Skills Requirement• To give students the opportunity

to apply mathematical and statistical reasoning in a chosen discipline.

• To involve students in the applicationof quantitative skills to problems that arise naturally in the discipline, in a way that advances the goals of the course, and in a manner that is not merely a rote application of a procedure.

Each QR applied course mustinclude at least two QR projects. A proj-ect might, for example, include datacollection, discussion of the data, col-laborative work on finding appropriateuses of the data, and use of appropriatetechnology in presentation and writing.The result of each QR project shouldbe a written assignment that includes astatement of the problem, an explana-tion of the methods used, and a

summary of the results. When appro-priate, the written assignment shoulddiscuss any limitations encounteredand possible improvements to the procedure and/or results.

In the first two years of the program,Hollins had thirty-seven QR appliedcourses from a variety of disciplines,including history, philosophy, theatre,classics, sociology, political science, economics, humanities (French), biology, chemistry, physics, mathematics,statistics, and computer science. TheHollins program was cited as one ofnine exemplary programs in the nationin Mathematics and Democracy.3

FA C U LT Y D E V E L O P M E N T

A major boost to the quantitative rea-soning program came when Professorsof Mathematics Caren Diefenderferand Patricia Hammer applied for andreceived an NSF Grant for “A FacultyDevelopment Program for QuantitativeReasoning Across the Curriculum.”This grant (NSF/DUE Project 9952807)brought four visiting scholars (JerryJohnson from the University of Nevadaat Reno, Dorothy Wallace fromDartmouth College, Helen Lang fromTrinity College in Connecticut, andLou Gross from the University ofTennessee at Knoxville) to Hollins during the 2000-01 school year. Eachscholar gave an evening lecture to theuniversity community and led a facultyworkshop to explain how he or sheapproached quantitative methods in a variety of disciplines.

Professors Diefenderfer andHammer led two series of four-dayworkshops in which faculty membersdiscussed the recently publishedMathematics and Democracy, investigated


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topics in Hollins’ basic QR course (AnIntroduction to Quantitative Reasoning),shared and critiqued one another’s QR project ideas, and presented theirQR work in progress. Thus, instructorshad the opportunity to test their assignments on a willing audience andto receive feedback on their proposedprojects. Twenty faculty members par-ticipated in the NSF-funded workshopsand their work resulted in twenty-sevenQR applied courses.

We were happy with both the number of participants and the variety

of disciplines represented (see AppendixC). Anita Solow, VPAA at Randolph-Macon Woman’s College from 1998 to2003, prepared an evaluation of theprogram and wrote, “The faculty unani-mously found the project to have valueto them. Although many of the com-ments were expected, I was impressedby the number of faculty members who explicitly talked about the facultydevelopment benefits to them andtheir sense of empowerment when itcame to quantitative reasoning.” As animportant side benefit, the workshopsprovided an opportunity to establishcross-disciplinary connections on campus and to increase discussion ofteaching beyond issues related to quantitative reasoning.

After the NSF workshops, facultymembers were invited to submit theircourses for approval as QR applied

courses. The application form is fairlysimple and requires instructors todescribe the two QR projects so that thereviewers can ascertain that the pro-posed projects fit the guidelines. From2001 to 2003, Professors Diefenderferand Hammer served as the screeningcommittee for these applications. Theyreviewed the applications, made theirrecommendations (and occasionallyasked faculty members to clarify orrevise certain ideas), and sent the applications to the Academic Policycommittee for final approval. Currently,

Professor Diefenderfer works with our director of quantitative reasoning,Professor Phyllis Mellinger (hired infall 2003), on this review. We hope toestablish a QR advisory board to serveas a screening committee to work withthe QR director in the future. Thiswould insure broader input from thefaculty members teaching QR appliedcourses and promote less dependenceon the mathematics department forprogram leadership.

Five faculty members who did notparticipate in the NSF funded work-shops are responsible for teachingtwelve of the thirty-seven approved QRapplied skills courses. This indicatesanother side benefit of the grant. Itgenerated enthusiasm for the programthat has encouraged additional facultymembers to participate and thus, thegrant lives on.

The workshops provided an opportunity to establish cross-disciplinary connections on

campus and to increase discussion of teaching beyond issues related to

quantitative reasoning.


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Participation in the workshops,and indeed in the QR applied programin general, is entirely voluntary for faculty. Professors who find themselvesworking on too many initiatives already,those who question their own compe-tence to teach quantitative techniques,and those who may have doubts aboutthe validity of teaching QR across thecurriculum simply choose not to signon. On the other hand, a number of those who did not understand whatquantitative reasoning might mean orwho were uncertain about the programhave had the chance to learn about QR and have become enthusiastic supporters of the requirement and ofthe educational process involved.

S I X A P P L I E D Q U A N T I TAT I V ER E A S O N I N G P R O J E C T S

The creativity of the quantitative projectsthat are now embedded in our curricu-lum is the major success of the quanti-tative reasoning program at Hollins.Here we highlight six projects todemonstrate both the variety of disci-plines and the meaningful applicationsthat characterize our program.Individual professors developed thesesix projects during on-campus NSF-funded workshops in January andApril/May 2001.

1. American Social History:Professor Ruth Alden DoanProject: Census of Families in Bristol, Rhode Island, in 1689In American social history, we focus on themes related to family, work, andcommunity. Assignments on PuritanNew England come at the very beginningof the semester, and therefore theseassignments introduce students to a

number of concepts, skills, and approach-es that we draw upon throughout thesemester. Goals of the QR project inparticular include:• To encourage students to think about

aspects of family life that change over time: structure, function, relationships within the family. This project focuses especially on structure.

• To learn some of the vocabulary usedin distinguishing types of families (nuclear, blended, and augmented, for example).

• To expose students to the kinds of data that historians might use and to encourage them to see both how richand how limited such data might be.

• To exercise certain skills in quantita-tive reasoning, including finding the mean, median, and mode of a data set, and creating meaningful tables or graphs.

We begin this unit of study by reading and discussing both primaryand secondary sources on Puritan NewEngland. The readings do not centeron families or on family structure, butrather provide a context for the Bristolcensus. Thus, for example, studentsread about and discuss Puritan behav-ioral norms, the physical shape of aPuritan town, and Puritan spirituality.We then turn to a series of handouts:vocabulary of family structures, theBristol census itself, and specific ques-tions about the Bristol census.

Quantitative exercises are dividedbetween those done in class and those done as homework. In-class workincludes counting the number ofhouseholds of each size and finding themean, median, and mode of householdsizes. Homework includes counting the number of children per householdand determining the mean, median,


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and mode of these data. Thus we prac-tice together the same kind of workthat students are sent off to do individ-ually. In class, we discuss representationsof the data. Does a pie chart work forthe data? A histogram? Why or why not?

The final product of this project isa five-page paper. Students write aboutwhat they have discovered in their min-ing of the Bristol census. They representtheir data in tables or graphs and refer to those representations of thedata in their texts. They are also askedto speculate about issues that might be implied rather than proven by the

census. Finally, they discuss the limita-tions of the data. What questionsremain that cannot be answered by the available information?

The students expressed a greatdeal of enthusiasm for this assignment,somewhat to my surprise. Ten out ofeleven in the first class I taught workedwillingly and enthusiastically. One stated that she was not good at mathand that she had math anxiety, and sheshowed some inclination to withdrawfrom group participation. (One studentconfessed after the fact that she was terrified but had chosen not to show it.)All, including the self-professed victim of math anxiety, did more thanadequate work in the end.

2. Ancient Art: Professor Christina A.Salowey Project: A Quantitative Analysis of Doric TemplesStudents in ancient art learn the basicelements of Greek architecture by studying both primary source texts andarchaeological data. The goals of thisspecific project are to introduce thetextual and material evidence for theDoric temple, to introduce quantitativeskills necessary for deeper understand-ing of ancient Greek architecture, and to show an application of theseskills that a working archaeologist would use in the field.

The creativity of the quantitative projectsthat are now embedded in our curriculum

is the major success of the quantitative reasoning program at Hollins.

Does the ancient world measure up? Students in theAncient Art course examine data from archaeologicalfindings and compare their computations of the ratio of column height to column diameter to see if they match the ideal of Vitruvius.


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At first students learn the preciseand detailed vocabulary that is necessaryto describe specific features of Greek architecture. Vitruvius’ de Architecturadefines in great detail the mathematicalrelationships between and proportionsof the elements of an ideal Doric temple. However, as we look at varioustemples from 600 to 250 BCE we discov-er that Vitruvius’ ideals are not alwaysmet. For example, when we use theexisting data and graph the ratio of col-umn height to column diameter duringthis time period, we discover that theratio varies a great deal. The graph ofthese ratios encourages students to raisemany questions and also shows thatthere are significant gaps in thearchaeological data, an indication thatbeckons students and professionals tocomplete future field work in templeanalysis.

As students become familiar with the ideal proportions of Vitruvius,we observe and measure a terracottatriglyph from Cosa (a temple fromsometime between the first centuryBCE to the first century CE) and recon-struct a temple plan based on this smallfragment. Students follow a hexastyleprostyle floor plan and are able to cal-culate the actual temple dimensions.We then draw a detailed scale diagramthat includes the exact number ofcolumns, column width, columnheight, and distance between columns.In creating this scale diagram we triedto convert all measurements to metersand had each square grid on our graphpaper represent one square meter.However, our calculations involved fractions of meters that were cumber-some to deal with and difficult to drawaccurately. We quickly discovered that itis much easier to use a module systemand assume that each square grid on

the graph paper is a square module.The amazing thing about this discoveryis that it became clear to us that theancient Greeks knew and understoodthis concept. Using the module systemmakes our drawing process much easier and our result more accurate. It was exciting to experience this intellectual moment of discovery andunderstanding with my students.

3. Calculus I: Professor Trish HammerProject: The McDonald’s Coffee Lawsuit—AKA, Newton’s Law of CoolingStudents in traditional Calculus I classes spend most of their time withformulas and algebraic manipulations.Calculus Reform of the 1990s emphasizesconcepts and applications. I stronglybelieve that students understand theconcepts of calculus by seeing how theyapply to real-world problems. The goalof this project is to show students aninteresting application of mathematicsin a field that most people woulddescribe as nonquantitative: law. Moststudents have heard about the McDonald’scoffee lawsuit. In fact, most have developed a nonmathematical opinionabout the case, which is very different from approaching a problem with no background or context. This projectallows students to view and analyze anewsworthy problem in a mathematicalway. The idea for this project comesfrom “Coffee to Go.” 4

In class, we discuss several basic dif-ferential equations and their solutions.In addition, we have studied Newton’slaw of cooling and have completedsome easy examples that deal with cool-ing objects and Newton’s law of cooling.The problems we work together in class consider only one environment.As students complete this project, they must take the coffee from one


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What do the golden arches of McDonald’s have to do with a seventeenth-century scientist? Calculus studentsuse Newton’s law of cooling to determine if McDonald’s is liable in the well-publicized coffee lawsuit.


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environment (a person’s hands) andmove it to a second environment (a cup holder).

Students work together duringnonclass times, in groups of two or three,to complete the assignment. I requirestudents to present both the mathemat-ical work and a verbal argument thatgives their verdict on whether McDonald’sis liable for damages.

In Calculus I, I was very pleased with students’ work. Students really getinto this project, possibly because many of them have a strong opinion on thecase beforehand. One group submitteda letter on law firm stationery statingtheir opinions. Another group submitteda very elegant mathematical solution,in which they used a single formula togive the temperature of the coffee afterso many minutes in the first environ-ment and so many minutes in the secondenvironment. The best solutions alwaysinvolve some discussion about “if theplaintiff was off by x seconds, then thecoffee was above (or below) the industry standard.”

Since I have used this assignmentfor several years, I’ve learned that Ineed to stress how I want students towrite up their solutions. I tell studentsthat they should think of this as a technical report that includes writingand mathematical calculations. I likethe fact that students feel that theyhave solved a real-world problem whenthey complete this project.

One of the great things about this project is that even after studentscomplete the mathematics, there isplenty of room for interpretation. What if the coffee is above (or below)the industry standard by 0.3 degrees?Should we round up, round down, or round at all? Is 0.3 degrees worth a million dollars? Is the possibility of

being off by 30 seconds a big deal? I get many different and correct inter-pretations that are based on the samemathematical calculations. It is interest-ing to see how students use their mathematical work to support theiropinions.

4. Ecology: Professor Renee D. GodardProject: Demography—Life Tables,Survivorship Curves, and Patterns ofFecundityPopulation biology covers a myriad ofsubjects: longevity, mortality, sex ratios,fecundity, and so on. The numbers can be daunting to students, and it isdifficult to collect meaningful data in a time period as short as one semester.The purpose of this project is to intro-duce students to concepts associatedwith understanding the dynamics of a population in a way that grabs theirattention. Focusing on the grave mark-ers of our own species avoids some ofthe usual pitfalls of population biology.

To prepare for the project, we read the textbook material relevant topopulation biology. Students also readthe materials handed out for the lab,and those materials include backgroundinformation. By the time this projectcomes along, the students are fairlycomfortable with working with data inExcel, so that part of the project doesnot have to be introduced. Before we proceed to the project itself, we talkabout questions raised by the readingsin class and about what we are going todo when we head out to the graveyard.

Working in pairs, the students collect data in the graveyard. In fact,because two lab sections carry out thisproject during the same semester, wecan gather data from two graveyards.Within each lab, the pairs collect theirdata, and then the whole group comes


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together—still in the graveyard—to hear summaries of the data. After thecollection of data, I pull it all togetherand hand it back to the students towork with. The summarized data theyreceive are essentially the beginnings oflife tables for the two sexes in the twocohorts. In class, we discuss how thestudents should complete the life tables(essentially what the variables mean),how to draw survivorship curves(including discussion of the value ofsemilog graphs), and how they might

approach the questions asked.Of course, the ideal approach to

population biology would be to follow a population from birth to death, but most of the inherently interestingspecies have lifespans that far exceed a semester. However, our Octobergraveyard walks are memorable to thestudents, and examinations of life-history patterns of our own speciesmakes the material come alive.

This was also a positive experiencefrom a faculty perspective. It gave methe opportunity to bring in currentfecundity data and apply it to the sur-vivorship patterns of the earlier group.The project illuminates how patterns of survivorship as well as birth rates and age at first reproduction influencepatterns of population. Thus a numberof significant points arise out of thisproject and dramatize important concepts.

5. France Since the Revolution:Professor Andre SpiesProject: A Quantitative Analysis ofNapoleon’s March to MoscowStudents of history know that Napoleon’smarch to Moscow, and the retreat of his forces, took place under cold andbrutal conditions. In this project studentswork with a nineteenth-century docu-ment that brings the disastrous historicalevent to life. As they consider suchquestions as what proportion of thoseordered to Moscow survived the marchto that city, how many soldiers died

A grave undertaking? Ecology students gather data in a graveyard in order to explore concepts inpopulation biology.

Our October graveyard walks are memorable to the students, and examinations

of life-history patterns of our own species makes the material come alive.


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per day on each stretch of the march,and, on average, how many yards laybetween corpses, students discover andexperience the conditions of Napoleon’smarch and retreat in vivid terms.

Students do not need advancepreparation for this assignment; rather,they work through the material togetherin class. I distribute copies of the map of Napoleon’s march and retreat,

along with copies of questions to beaddressed. Students take turns standingat the board to record the estimatesand conclusions of the group. Somestudents measure distances on the map.Other students wield their calculatorsto translate numbers and distances intoanswers to the questions. An essentialpiece of this group work is that I’m a participant and not an expert. I workthrough the material with the studentsrather than preparing it ahead of time,and students see me as a colleague in search of interpretations.

After the group works through thequestions in class, students write papersindividually. Their papers must includean explanation of the mathematicalprocesses. Conclusions must be presentedboth in numerical form and in prose.Students not only analyze the resultsbut also discuss what they have learnedthrough the project. They are asked, in fact, to include adjectives that theywould use to describe Napoleon’s march.

Students did very good work onthis project. Seven out of eight in thefirst class to tackle it bought into it.Even students who said that they had

math phobias succeeded—sometimesdid the best work, in fact.

The benefits of the QR component,and of this project, are many. The in-class work loosened up the class and so contributed to improved class dynamicsthroughout the semester. I becamereacquainted with quantitative methodsto the point that I am now studyingregression analysis, a skill that I hope

to use in my own research. For bothstudents and myself, this project provedbroadening in the way that the liberalarts are supposed to be.

The QR project changes the waysthat students see the study of history.This approach demonstrates to stu-dents that professional historians havea variety of interests and skills, that we take interdisciplinary work seriously,and that history professors are notafraid to try some different approachesand different ways of examining materi-al. Similarly, the QR project exemplifiessome of the key virtues of the generaleducation program: it shows that manydisciplines draw and depend on a num-ber of different skills—even historianscan do quantitative work!—and QRprovides a model of how the teaching ofskills across the curriculum works well.

6. Lighting Design: Professor Laurie Powell-WardProject: Creating Even Washes of Light on a Theatrical StageIn the Lighting Design class studentsexamine the potentials and problemsof theatrical lighting through lab

In this project students work with a nineteenth-century document that brings

Napoleon’s march to Moscow to life.


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exploration with standard industryequipment. Students work with script-based design projects and develop the technical support information thatis necessary to produce a stage design.One of the QR projects is designed to allow students to use specific lightingequipment in the Hollins theatre toproduce even washes of light on thestage. The specific goals of the project are:

• To create a light plot that includes three washes for a bare stage.

• To choose correct instrumentation.• To determine the best placement

of instruments for continuity of angle of light.

Students begin by breaking downthe stage space (via a ground plan) into sized areas. These sized areas arecircles, each of which represents thecircular area on the stage that can beilluminated by an individual light. We have a group lab session to becomefamiliar with the instruments and thequality of light they produce. In addi-tion, I work with students to developvocabulary, introduce the ground plansand section views that we’ll be complet-ing, and understand how to use scalemeasurements.

The assignment requires studentsto prepare three different washes oflight. We complete the first wash togetheras a group. Students then complete the second and third wash on their own,using the group work as their model.Classes consist of ten students because

of limited lab space, and this size makesit easy and fun to work as a group in class on the technical paperwork.

I was pleased with the students’work on this project. At first, many wereoverwhelmed by the number of stepsinvolved in making decisions. However,part of the exercise is learning how tofocus on one step at a time, even thoughthe task seems chaotic. By the end of

the project, students were comfortablewith the process.

The first time I used this project, I introduced it as our QR project andnoticed that many of the studentstensed up for a full class period beforethey relaxed and realized that theywere able to understand and completethe task. The next time I teach thisclass, I will simply introduce this as a lighting project and hope that approachwill create less anxiety. In addition,when I assign this project again, I willuse a smaller stage space and include a smaller number of lighting areas inorder to reduce the frustrations thatstudents experience at the beginning of the task. I will also have the classwork on the project in the lighting labin order to have a three-dimensionalreference instead of relying on the two-dimensional paper representation.I plan to continue to use the entiresequence of three assignments, becausethis was effective and allowed studentsto master one component at a time asthey build on the previous assignment.

My students have discovered that a theatre design class is not an easy A or a

filler course, but one that is intellectually challenging and artistic at the same time.


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While completing this project, mystudents became aware of the manysteps involved in creating a lightingdesign for stage. They now appreciatethe systematic and structured nature of lighting design in the early phases, a process that must occur before the“artsy” work in the theatre space. Thisproject destroys the myth that all lightinginstruments are the same and that any one will work fine if you just pointit in the right direction. My studentshave discovered that a theatre designclass is not an easy A or a filler course,but one that is intellectually challengingand artistic at the same time. I am pleased that my students now believethat applied QR is not necessarily evil.

This class worked so well that I haverecently created QR projects for myScene Painting course, and it has been approved as one of the QR applied courses. I continue to examine my otherdesign courses and am discovering that quantitative methods are inherentin everything I teach.

Short Descriptions of AdditionalProjectsComputer Science I: Professors DanChrisman and James AllenGiven a very large set of data, what’sthe quickest way to compute the meanand standard deviation? Students inComputer Science I develop and thenanalyze both one-pass and two-passalgorithms to determine the bestmethod.

Economics of Health Care: Professor Juergen FleckAre lifestyle choices as important indetermining a person’s health status asaccess to health care? Students analyzethe effects and social costs of particularbehaviors, including alcohol consump-

tion, drinking and driving, speeding,inadequate nutrition, lack of exercise,smoking, and drug use.

Human Memory: Professor George LedgerHow can we statistically analyze, describe,and interpret the results of a memoryexperiment? Students work with thedata from the Ledger and Mays (1994)metamemory study. They identify nominal, ordinal and ratio variables.They compute means for all measuresby graduating class and create bar chartsto describe each. They create scatter-plots by class and create new categoriesbased on the distribution of scores.They compute correlations among thevariables, using both pair-wise and list-wise deletion of missing data andinterpret the results. Throughout thecourse they are asked to identify theadvantages and disadvantages of eachre-categorization and analysis and toprovide justifications for their decisionsand interpretations of the results.

Introduction to Statistics: Professor Julie M. ClarkStudents complete three QR projectsfor this course. In the first project, they practice collecting, organizing,and interpreting data of their ownchoosing. The second project involvesperforming a simple linear regression—finding data for two variables that theysuspect might be related, and thenfinding a linear relationship betweenthe variables and calculating the corresponding fit of the model. Studentshave chosen to study such diverse topics as brain size vs. IQ, diamondprices vs. carat weights, and public schoolexpenditures vs. SAT scores by state. In the third project, students critique a published report of either a statisticalsignificance test or estimated population


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value (confidence interval). With all projects, students choose their own topics and find or collect their owndata so the projects are meaningful,interesting, and relevant to their individual disciplines.

Methods of Matrices and Linear Algebra:Professor Caren L. DiefenderferLinear algebra is an important subjectbecause of its applicability to many disciplines. For each of the QR projectsin this course, students may choose one from a list of roughly ten applica-tions. In this way, students can createassignments to match their personalinterests. Science students opt for thechemistry, biology, and physics applica-tions; social science students choose the economics, business, and planningtopics; computer science students enjoythe computer graphics projects; andother students usually prefer to selectapplications from a variety of fields.Cryptology (creating and breaking codes),Markov Chains, and the AssignmentProblem are three of the all-timefavorite projects.

Physical Principles and Analytic Physics:Professor Joseph AmetepeEdwin Hubble considered the relation-ship between distance from the Earthand the recession velocity of galaxies,because he hoped this would shed somelight on how the universe was formedand what we may expect to happen in the future. Physics students useHubble’s data to find the correlationbetween distance and recession velocity.In addition, they use these data tomake predictions about new galaxies.Students also compare Hubble’s law tothe regression equation and explainwhether the two approaches give consistent or inconsistent information.

Research Methods in Communication:Professors Jane Tumas-Serna and Chris RichterWhen is the news truly news and whenis it entertainment? Does televisionnews reflect gender biases? Students inthis communication studies courseaddress questions like these when theydo a content analysis of the TV news.Students record one week of networkor network affiliate news programs forABC, NBC, and CBS. They devise con-tent coding schemes and then code theprograms for such things as amount of time devoted to hard, soft, or tabloidnews or amount and type of news covered by male and female reporters.Finally, students do statistical analysesto compare the categories across thedifferent stations and to determine theirlevels of intercoder reliability.

Research Methods in Political Science:Professor Jong O. RaIs there a relationship between a voter’s(perceived) financial condition and herpresidential vote? Students learn howto use a three-way table among threevariables (the voter’s perception of herown financial condition, the shape ofthe national economy, and her evalua-tion of the proposed management of these issues by Bush and Gore in the2000 presidential election) to answerthis question.

Sociology of Health, Illness, and Medicine:Professor Kay R. BroschartDo nations that spend more money on health care have healthier citizens?Is there a connection between the proportion of women physicians in a medical specialty and their averagesalaries? Students discover how to analyze data and give informed answersto these questions.


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Symbolic Logic: Professor Michael GettingsIs symbolic logic useful in today’s society? Students in Symbolic Logiclearn how to apply everyday reasoningprocesses such as argument, inference,premise, and conclusion as they analyzenonfiction and newspaper articles in ordinary language. Students look at specific excerpts from WashingtonPost editorials and from John KennethGalbraith’s The Affluent Society. Over several years of teaching, ProfessorGettings has seen that the textbookexamples are often contrived, but articles from other sources have hiddenassumptions and prove to be morechallenging and beneficial to students’understanding. In-depth understandingalso helps students to retain importantprinciples.

P E D A G O G Y

A change in the general educationrequirements at any institution affectsboth students and professors, sometimesin unexpected ways. A reconsiderationof general education requirementsshould cause each professor to reflecton the material contained in a givencourse topic, how it is presented to thestudents, and what the students areexpected to gain from the educationalexperience. The implementation of an applied quantitative reasoningrequirement in the general educationprogram at Hollins encouraged profes-sors from many different disciplines

to recognize that quantitative reasoningwas a skill necessary for the mastery of material in courses they were alreadyteaching. Faculty who teach quantita-tive reasoning material in their coursesobserve a variety of benefits both totheir development as more active,engaged, and creative teachers and tothe enhancement of their students’classroom experience.

The NSF-funded seminars were aproductive first step for many profes-sors. The introductory readings andinteractions with the organizers,Professors Diefenderfer and Hammer,gave the participants ideas aboutchanging their own classes to includequantitative reasoning assignments.The seminars also provided a templatefor writing a quantitative reasoning

assignment. For faculty who teach infields other than mathematics and the sciences, the creation of an exercisethat guides the students step-by-stepthrough a problem-solving process maynot be a regular part of their classroompreparation. The peer review of thequantitative reasoning assignmentsproved instructive for all participantsand added to their pedagogical reper-toire. The gathering of faculty from several different disciplines had theadditional benefit of increasing colle-giality among those faculty, generatinginterest in the goals of courses taught

The QR requirement at Hollins serves as a cornerstone of the skills aspect of the

new general education program and is a model for inspiring more effective

teaching and learning.


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outside of their own disciplines, andraising the level of respect they had foreach other’s intellect, scholarly ability,and pedagogical technique. The semi-nars thus expanded the pedagogicalhorizons of the faculty involved and created a mini-community of explorationand learning that bolsters the educa-tional mission of the university.

Many faculty noted that the techniquesthey used to present the quantitativereasoning assignments created new

classroom dynamics. A number of theassignments were multistep proceduresinvolving data collection, calculation,presentation of the results, and inter-pretation. The step-by-step nature ofthe assignments alleviated student anxi-ety about the quantitative component,made the assignments accessible, and had a natural fit with the coursematerial. Group work, especially fordata collection, was a common feature,and faculty found that students enjoyed

working with each other. Students discovered that they each had differentstrengths, which contributed to solvinga complex problem. Student collabora-tion added a strong element of fun andexcitement to the task, thus removingany stigma that the quantitative reasoningcourse label might have carried. Insteadof being daunted by the in-depth problem solving they were being askedto do, students became engrossed in the work, drawn in by the practicalnature, and proud of the results theywere able to produce.

The quantitative reasoning exercisesdeveloped by many faculty membershave the added benefit of coming fromreal-life examples and data. In ProfessorDoan’s history class, students wrestlewith how to deal with the individualslisted in the Bristol census who do notfit into neat family units. In ProfessorSalowey’s classics class, the measurementsof extant Doric temples raise manyopen-ended questions about templesfrom antiquity: Was Vitruvius’ ratioactually the standard, or was it only hisimagined ideal? In addition, ProfessorSalowey’s project on the ratio of col-umn height to column diameter showsclearly that professionals need to findand analyze more data from the timeperiod of Vitruvius. In Professor Spies’history class, as students investigate theextraordinary document that recordsquantitative information about Napoleon’smarch to Moscow, their analysis conveysa seriousness and weight that make this historic event become real, present,and disastrous. In each of these settings,academic material comes to life whendealing with the messy, complicated,sometimes incomplete real-life data. We have discovered a powerful principle:using data from real situations invitesstudents to explore and even struggle to

Faculty members re-experience the joys of being astudent. During NSF-funded faculty developmentworkshops, we shared and critiqued one another’sideas and enjoyed learning about quantitativeprojects in an amazing variety of settings.


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understand the material so that theyown the process of interpretation.Contrived textbook examples are simpleand easy, but do little to build confidenceor promote deep understanding.

The quantitative reasoning require-ment at Hollins serves as a cornerstoneof the skills aspect of the new general education program and has proved tobe a model for inspiring more effectiveteaching and learning. The assignmentsencourage many faculty to move in new pedagogical directions, providingbenefits for their students not only ingeneral education but also in the mas-tery of knowledge and skills essential to their specific fields. The quantitativereasoning program invites students to use all the resources of the academiccommunity, an important hallmark of a successful liberal arts education.Barriers between disciplines begin to fade when a student in a history classthinks about the meaning of numbersreported in statistical documents orwhen a student in a calculus class notesthe importance of the interpretation of a liquid cooling curve in a multimil-lion-dollar lawsuit. Students will leavethis academic environment not onlyknowing how to read texts and images,but also how to read the important language of numbers and be critical of their interpretations.

A S S E S S M E N T

We have used several assessment instruments to measure the effectivenessof the Hollins program since we introduced our basic skills quantitativereasoning requirement in the fall of1998. One measure of effectiveness isto consider final course grades. Of the363 students who enrolled in Mathematics

100 during the past five years, 85.1%passed the course. Of those who completed the course, 88.5% passed.(Fourteen students withdrew from the course within the approved drop/add period.) The 302 students whoenrolled in this course and received agrade earned an average of 2.18 qualitypoints. (On the Hollins scale, a C receives2 quality points and a C+ would receive2.3 quality points.)

All entering students complete a quantitative reasoning assessment and students enrolled in the basic skills quantitative reasoning class retake theassessment at the end of the semester.A comparison of these two scores givesus another measure of their progress.Our data from this five-year periodshow that on average, students improve13 percent on the assessment aftercompleting the course. (The assessmentis based on 50 points and the averagescore improvement is 6.5 points.) Bothof these numerical measures indicatethat students who enroll in the coursebecome stronger in analyzing quantita-tive information.

We have also collected qualitativedata. During the 2001-02 year, a campusassessment committee devised studentperception surveys to assess the newgeneral education requirements. Thedata for students in the basic skillsquantitative reasoning class show a veryhigh self-assessment of improvement.The percent changes range from +93.9percent to +140.7 percent on four surveyitems. Data for students in the appliedskills quantitative reasoning classes alsoshow positive student self-assessment,with percent changes of +51 percent to +52 percent.

Hollins has been selected to be apilot site for an NSF grant, “TheDevelopment of Assessment Instruments


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for the Study of Quantitative Literacy,”under the direction of Susan Ganter at Clemson and Jack Bookman at Duke.The purpose of this grant is to developappropriate assessment instruments for quantitative literacy and to train asmall group of professionals to conductnationwide assessments. Hollins facultymembers will help to write the assess-ment instruments and then test theinstruments on campus. This work willoccur during 2004.

Since the program is relatively new, we do not yet have feedback fromour graduates describing whether skillslearned in their QR courses at Hollinshave been beneficial to their lives after graduation. A future evaluation will survey alumnae. Hollins faculty membershave been invited to give keynoteaddresses, lead workshops, and presentpanel discussions at professional meet-ings. The number and variety of theseinvitations document the admirationand high esteem that professionals across the country have for the HollinsQR program.

C O N C L U S I O N S

There are many practical reasons forincluding courses in quantitative rea-soning in an undergraduate curriculum.The skills learned in these courses will help students in both their futureacademic work and the job market.Prospective employers are often impressedwith candidates who have strong quan-titative skills. Informed citizens need

to know how to make arguments andinterpret data they see and hear in thenews. Learning how to make appropriatechoices when faced with complex issuesis an important life skill.

In addition, there are strong academic reasons for preparing studentsto think quantitatively. Techniqueslearned in our basic and applied coursesgive students the confidence to solveproblems in new situations. Facultymembers at Hollins believe that theproject-oriented nature of the appliedquantitative reasoning assignmentsencourages students to approach discovery and interpretation as profes-sionals. In short, students are gettingmore practice in solving real-life, open-ended problems.

Providing all students with theknowledge, ability, and confidence tosolve and understand quantitative issueswas the original goal of the Hollinsquantitative reasoning program. Indesigning projects to give students theseskills, faculty members have becomemore conscious and deliberate aboutcreating assignments in all their classes.We have discovered that quantitativeperspectives can be found everywhere,in many disciplines and in many classes,not just in the classes that now satisfyour applied quantitative reasoningrequirement. Our collective efforts tocreate projects and assignments thatwill strengthen the quantitative reason-ing abilities of our students are teachingus a great deal about interdisciplinarywork. The journey continues to beenlightening for us and for our students.

1 Mathematics and Democracy: The Case for Quantitative Literacy, edited by Lynn Arthur Steen, prepared by the National Council on Education and the Disciplines, 2001, p. 2.

2 Mathematics and Democracy, p. 6.3 Mathematics and Democracy, p. 114.4 Instructor’s Resource Manual for Calculus from Graphical, Numerical, and Symbolic Points of View, Volume 1,

Saunders College Publishing, 1995, p. 106.


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A P P E N D I C E S

Appendix A A History of Quantitative Reasoningat Hollins, 1996–99For some time, Hollins had recognizedweaknesses in the general quantitativeskills of its students. In 1996, after acareful reading of the MAA report,Quantitative Reasoning for College Literacy,Caren Diefenderfer, associate professorof mathematics, and Ruth Doan, professor of history, submitted a pro-posal that “Hollins shall institute aQuantitative Reasoning requirementthat will draw upon courses across the curriculum and will be supportedby a Quantitative Reasoning Center.”Support for this proposal gainedmomentum early in 1998 when HollinsUniversity was required to “demon-strate that its graduates are competentin…oral communication, fundamentalmathematical skills, and the basic useof computers” during a reaccreditationstudy by the Southern Association ofColleges and Schools (SACS). At thistime, professor Diefenderfer and PatriciaHammer, associate professor of mathe-matics at Hollins, secured from facultyand senior administration a commitmentto the importance of quantitative rea-soning across the curriculum. In spring1998, Hollins began to develop a QRprogram similar to that at WellesleyCollege. The Hollins program not onlysatisfies the SACS criteria, but alsoemphasizes quantitative reasoning acrossthe curriculum and gives students the opportunity to learn the broad significance and applicability of quanti-tative reasoning. A two-part quantitativereasoning requirement consisting of a QR basic skills requirement and a QR applied skills requirement wasapproved by the faculty in spring 1998,

with the basic skills requirement tobegin in the fall of 1998 and both thebasic and applied skills requirements tobe in place by fall 1999. During 1998-99, the mathematics and statisticsdepartment under the leadership ofProfessors Diefenderfer and Hammerworked closely with Hollins faculty todefine, identify, and more fully developthe guidelines for QR applied skillscourses.

Since the quantitative reasoningrequirements outlined in 1998 fit per-fectly into the framework of the newESP program and paralleled the role ofwriting and oral communication acrossthe curriculum, we decided thatinstead of following the originaltimetable and implementing the QRapplied skills requirement in the fall of1999, we should wholeheartedly sup-port this new curriculum and imple-ment the second QR requirement atthe same time as the new ESP program.While this delayed the implementationof the QR applied skills requirementfor several years, it also gave us time towork with interested faculty membersand develop some high-quality projects.

Professors Diefenderfer andHammer received $3,000 from theHollins University Sowell Fund for faculty development during the springsemester of 1999. They led a series ofworkshops to encourage Hollins facultymembers to incorporate quantitativereasoning into existing courses. Theworkshop met four times and had tenparticipants. These participants workedtogether to create a Hollins definitionof quantitative reasoning (see AppendixB) and wrote guidelines concerningdevelopment of QR modules and classification of courses as QR applied.In addition, each participant createdone or two QR projects that they used


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and revised during the subsequent academic year. While these faculty workshops were successful in promotingboth camaraderie and the ideas ofquantitative reasoning, campus-wideawareness and understanding of quantitative reasoning still fell short.Nine of the ten faculty members whoparticipated in these activities werefrom science, social science, and mathematics departments.

Professors Diefenderfer andHammer received another SowellGrant ($3,000) during 1999-2000 tocontinue their efforts. Since we wantedto attract faculty members from thehumanities and the arts, as well as thesciences and social sciences, we pro-posed a faculty reading group, with thehope that this format would appeal to a broader cross section of our facultyand that the discussions would help to promote a deeper understanding ofquantitative reasoning on our campus.

We had approximately twenty facultymembers participate, we read WhyNumbers Count—Quantitative Literacy forTomorrow’s America (published by theCollege Board, 1997), and we increasedthe number and variety of representeddisciplines. Meanwhile, discussions and deliberations on the new ESP program continued.

Appendix BThe Hollins Definition ofQuantitative ReasoningQuantitative reasoning is the applica-tion of mathematical concepts andskills to solve real-world problems. In order to perform effectively as professionals and citizens, studentsmust become competent in readingand using quantitative data, in understanding quantitative evidence,and in applying basic quantitative skills to the solution or real-life problems.


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Appendix CParticipants in the 2000–01 NSF-Funded QR Workshops at Hollins

Faculty Member Department Q course(s)

Joseph Ametepe Physics Physical PrinciplesAnalytical Physics

Rebecca Beach Biology Genetics*

Sandy Boatman Chemistry Biology of Women*

Kay R. Broschart Sociology Sociology of Health, Illness, and MedicineMethods of Social Research

Dan Chrisman Computer Science Computer Science I

Julie M. Clark Mathematics and Introduction to StatisticsStatistics Statistical Methods

Dan Derringer Chemistry General ChemistryPrinciples of Chemistry

Caren L. Diefenderfer Mathematics and Methods of Matrices and Linear Statistics Algebra

Ruth Alden Doan History U.S. Social History

Juergen Fleck Economics Economics of Social IssuesEconomics of Health Care

Sally Garber Mathematics and Intuitive CalculusStatistics

Michael Gettings Philosophy and Symbolic LogicReligion

Renee D. Godard Biology Ecology

Trish Hammer Mathematics and PrecalculusStatistics Calculus I and II

Nancy Healy Computer Science Microcomputers in the Business World*

George Ledger Psychology Human Memory

Laurie Powell-Ward Theatre Lighting Design

Jong O. Ra Political Science Research Methods in Political Science

Christina A. Salowey Classics/Art Ancient Art

Andre Spies History France Since the Revolution

Robin Taylor Biology Plants and PeoplePlant Biology

Jane Tumas-Serna Communication Research Methods inStudies Communication

*course is not yet approved

Caren L. Diefenderfer, associate professor of mathematics, earned M.A. and Ph.D. degrees from the University of California–Santa Barbara.Over the years her scholarly interests have focused on approximation theory, graph theory, and computer graphics. Her current work involvescombinatorics and the geometry of weaving. Professor Diefenderfer works with the College Board as chief reader and a member of the test development committee for AP calculus. She is the chair-elect of the newly formed Special Interest Group in Quantitative Literacy

of the Mathematical Association of America.


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Ruth Alden Doan, professor of history, received an M.A. and Ph.D. from the University of North Carolina–Chapel Hill. Professor Doan primarily teaches courses in early America and in social and religious history. She also teaches the survey of U.S. history, seminars on colonial history, the American Revolution, antebellum America, and the American wilderness experience. Her publications include a monograph on Millerism in nineteenth-century American religious history. Currently, she is working on a study of southern evangelicals

and narratives of religious experience.

Christina A. Salowey, associate professor of classical studies, has a Ph.D. from Bryn Mawr College, an M.A. from Tufts University, and an M.S. from Rensselaer Polytechnic Institute. Professor Salowey specializes in Greek art and archaeology and religion. She is interested in pedagogical issues and serves on the executive committee of the American School of Classical Studies at Athens. She codirects an archaeological survey in Arcadia, bicycling to all her sites, and continuesextensive research on the cult and labors of Herakles.

Holl ins Univers i ty • P.O. Box 9706, Roanoke, Virginia 24020 • www.hol l ins .edu

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