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Journal of Education for Business

ISSN: 0883-2323 (Print) 1940-3356 (Online) Journal homepage: http://www.tandfonline.com/loi/vjeb20

Quantitative Literacy for Undergraduate Business

Students in the 21st Century

Richard McClure & Sumit Sircar

To cite this article: Richard McClure & Sumit Sircar (2008) Quantitative Literacy for

Undergraduate Business Students in the 21st Century, Journal of Education for Business, 83:6, 369-374, DOI: 10.3200/JOEB.83.6.369-374

To link to this article: http://dx.doi.org/10.3200/JOEB.83.6.369-374

Published online: 07 Aug 2010.

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he฀ environment฀ in฀ which฀ busi-ness฀ enterprises฀ operate฀ today฀ is฀ radically฀different฀from฀that฀of฀previous฀ decades,฀ requiring฀ a฀ reassessment฀ of฀ how฀undergraduates฀in฀business฀schools฀ are฀ taught.฀ This฀ environment฀ has฀ been฀ shaped฀ by฀ deregulation,฀ globalization,฀ and฀the฀Internet,฀which฀have฀combined฀ to฀produce฀an฀intensely฀competitive฀situ- ation฀in฀which฀companies฀generally฀pro-duce฀ similar฀ products฀ and฀ have฀ access฀ to฀similar฀technologies.฀Therefore,฀com-panies฀must฀compete฀by฀differentiating฀ their฀ business฀ processes,฀ requiring฀ the฀ widespread฀use฀of฀business฀analytics฀for฀ effectiveness฀(Davenport,฀2006;฀Daven-port฀&฀Harris,฀2007).

The฀central฀theme฀of฀this฀article฀is฀that฀ quantitative฀ methods฀ can฀ and฀ should฀ be฀ applied฀to฀a฀wide฀array฀of฀decision-mak-ing฀scenarios฀and฀that฀all฀business฀students฀ should฀have฀an฀adequate฀level฀of฀quantita-tive฀literacy฀to฀make฀calculated฀decisions฀ in฀the฀increasingly฀complex,฀information-oriented,฀ knowledge-based฀ world.฀ We฀ subscribe฀ to฀ the฀ definition฀ of฀ quantita-tive฀literacy฀adopted฀by฀the฀International฀ Life฀Skills฀Survey฀(Dingwall,฀2000):฀“An฀ aggregate฀ of฀ skills,฀ knowledge,฀ beliefs,฀ dispositions,฀ habits฀ of฀ mind,฀ communi-cation฀ capabilities,฀ and฀ problem฀ solving฀ skills฀that฀people฀need฀in฀order฀to฀engage฀ effectively฀ in฀ quantitative฀ situations฀ aris-ing฀in฀life฀and฀work”฀(p.฀147).฀

Although฀ the฀ term฀quantitative฀ liter-acy฀ is฀ a฀ superset฀ of฀ the฀ term฀numeracy฀

(Lange,฀2003),฀we฀use฀them฀interchange-ably.฀ We฀ strongly฀ believe฀ that฀ numer-acy฀ relates฀ to฀ numbers฀ exactly฀ as฀ lit-eracy฀relates฀to฀words.฀College฀education฀ should฀stress฀the฀two฀equally,฀but฀such฀an฀ equal฀stress฀does฀not฀occur฀at฀most฀insti-tutions.฀Unfortunately,฀numeracy฀is฀often฀ mistakenly฀ equated฀ with฀ mathematics.฀ Instead,฀ it฀ is฀ more฀ of฀ an฀ approach฀ to฀ solving฀ problems฀ and฀ a฀ state฀ of฀ mind.฀ Students฀ cannot฀ achieve฀ numeracy฀ by฀ taking฀ more฀ courses฀ in฀ the฀ mathemat-ics฀ department฀ any฀ more฀ than฀ educa-tors฀can฀achieve฀literacy฀by฀adding฀more฀ courses฀ in฀ English฀ literature.฀ The฀ focus฀ on฀ quantitative฀ literacy฀ needs฀ to฀ be฀ in฀ every฀course฀in฀every฀department,฀just฀as฀ it฀should฀be฀for฀literacy.฀Steen฀(2004)฀and฀ Richardson฀ and฀ McCallum฀ (2004)฀ have฀ made฀the฀same฀arguments.฀

Although฀ business฀ schools฀ teach฀ how฀ swiftly฀ the฀ business฀ environment฀ is฀ changing,฀ instruction฀ in฀ quantitative฀ methods฀ has฀ barely฀ changed฀ in฀ almost฀ half฀a฀century.฀Academic฀institutions฀are฀ exceedingly฀ reluctant฀ to฀ change฀ their฀ curricula฀in฀quantum฀leaps฀(Bok,฀2005).฀ Major฀ external฀ forces฀ are฀ necessary฀ to฀ bring฀ about฀ such฀ change.฀ We฀ believe฀ that฀these฀forces฀are฀the฀changing฀nature฀ of฀ business;฀ the฀ loss฀ of฀ U.S.฀ competi- tiveness฀(only฀6฀of฀the฀top฀25฀informa-tion฀ technology฀ companies฀ are฀ based฀ in฀the฀United฀States);฀globalization฀and฀ outsourcing฀ to฀ foreign฀ countries;฀ the฀ threat฀of฀India,฀China,฀and฀South฀Korea฀

Quantitative฀Literacy฀for฀Undergraduate฀

Business฀Students฀in฀the฀21st฀Century

RICHARD฀McCLURE SUMIT฀SIRCAR MIAMI฀UNIVERSITY OXFORD,฀OHIO฀

T

ABSTRACT. The฀current฀business฀

environment฀is฀awash฀in฀vast฀amounts฀of฀ data฀that฀ongoing฀transactions฀continually฀ generate.฀Leading-edge฀corporations฀are฀ using฀business฀analytics฀to฀achieve฀com-petitive฀advantage.฀However,฀educators฀are฀ not฀adequately฀preparing฀business฀school฀ students฀in฀quantitative฀methods฀to฀meet฀ this฀challenge.฀For฀more฀than฀half฀a฀century,฀ business฀schools฀have฀relied฀mostly฀on฀a฀ course฀in฀calculus฀and฀a฀course฀in฀statistics฀ to฀meet฀the฀needs฀of฀their฀students฀despite฀ an฀information-based฀business฀climate฀that฀ has฀changed฀significantly.฀The฀authors฀pro-pose฀that฀educators฀prepare฀students฀in฀the฀ areas฀of฀mathematical฀modeling฀and฀risk฀ management฀and฀quantitative฀skills,฀teach- ing฀them฀in฀the฀context฀of฀meaningful฀busi-ness฀problems.

Keywords:฀business฀students,฀mathematical฀ modeling,฀quantitative฀literacy

Copyright฀©฀2008฀Heldref฀Publications

VIEWPOINT

as฀ major฀ economic฀ powers฀ (14฀ of฀ the฀ world’s฀ top฀ 25฀ information฀ technology฀ companies฀ are฀ based฀ in฀Asia);฀ and฀ the฀ emergence฀of฀a฀knowledge-based฀econ-omy฀in฀which฀82%฀of฀the฀workforce฀is฀ in฀the฀service฀sector.฀

From฀ our฀ discussion฀ with฀ faculty฀ in฀ the฀ present฀ study,฀ generally฀ faculty฀ resist฀ increasing฀ the฀ quantitative฀ liter-acy฀ of฀ business฀ students฀ because฀ they฀ believe฀that฀(a)฀all฀business฀students฀do฀ it฀ being฀ practically฀ nonexistent฀ in฀ busi- ness฀curricula฀(Kolata,฀1997).฀Our฀objec-tive฀ for฀ this฀ article฀ is฀ to฀ argue฀ that฀ to฀ compete฀globally฀and฀prepare฀American฀ business฀students฀for฀the฀future,฀the฀fol-lowing฀ are฀ necessary:฀ (a)฀ the฀ increased฀ use฀of฀quantitative฀methods฀in฀the฀core฀of฀ the฀undergraduate฀business฀program฀(i.e.,฀ the฀required฀courses);฀(b)฀a฀modification฀ of฀the฀quantitative฀tools฀covered฀to฀meet฀ emerging฀ requirements฀ in฀ business;฀ and฀ (c)฀ the฀ use฀ of฀ sophisticated฀ computer฀ software,฀ now฀ commonly฀ available฀ to฀ all฀organizations,฀to฀make฀even฀complex฀ computations฀ relatively฀ straightforward฀ for฀the฀ordinary฀manager.฀In฀the฀next฀sec-tions,฀ we฀ describe฀ the฀ emerging฀ impact฀ of฀quantitative฀methods฀in฀business,฀high-light฀ the฀ low฀ standard฀ of฀ mathematics฀ education฀in฀U.S.฀high฀schools,฀and฀show฀ that฀even฀selective฀business฀schools฀have฀ been฀affected.฀We฀then฀demonstrate฀that฀ the฀ quantitative฀ methods฀ courses฀ now฀ being฀taught฀at฀selected฀business฀under-graduate฀ programs฀ are฀ inadequate฀ and฀ that฀ the฀ current฀ business฀ environment฀ requires฀ increased฀ quantitative฀ literacy฀ on฀ the฀ part฀ of฀ all฀ managers.฀ Last,฀ we฀ make฀ recommendations฀ for฀ appropriate฀ course฀work฀to฀meet฀these฀needs.

The฀Future฀Is฀Now

After฀transforming฀science฀and฀engi-neering,฀mathematics฀has฀been฀steadily฀ transforming฀ many฀ fields฀ of฀ business.฀ Mathematics฀ transformed฀ finance฀ and฀ is฀now฀changing฀the฀conduct฀of฀a฀wide฀ array฀ of฀ (hitherto฀ untouched)฀ business฀

activities,฀ ranging฀ from฀ advertising฀ campaigns฀ and฀ newsroom฀ research฀ to฀ the฀ building฀ of฀ customer฀ relationships฀ (Baker,฀ 2006).฀ It฀ is฀ likely฀ that฀ faculty฀ members฀resisting฀the฀use฀of฀quantitative฀ techniques฀are฀not฀aware฀of฀these฀recent฀ developments฀in฀industry฀and฀that฀some฀ of฀those฀faculty฀were฀probably฀educated฀ when฀ mathematical฀ approaches฀ were฀ not฀ used.฀ The฀ situation฀ is฀ not฀ unlike฀ the฀ rapid฀ intrusion฀ of฀ computer฀ graph-ics฀ into฀ advertising,฀ which฀ essentially฀ rendered฀a฀large฀number฀of฀conventional฀ commercial฀artists฀obsolete.

In฀a฀recent฀study฀of฀32฀organizations฀ that฀had฀committed฀to฀quantitative,฀fact-based฀analysis,฀Davenport฀(2006)฀found฀ that฀ virtually฀ all฀ were฀ leaders฀ in฀ their฀ fields.฀ They฀ emphasized฀ business฀ ana- lytics฀as฀an฀overarching฀strategy฀cham-cal฀skills฀but฀a฀lot฀of฀people฀with฀the฀very฀ best฀analytical฀skills—and฀consider฀them฀ a฀key฀to฀your฀success.

2.฀You฀ not฀ only฀ employ฀ analytics฀ in฀ almost฀every฀function฀and฀department฀but฀ also฀consider฀it฀so฀strategically฀important฀ that฀you฀manage฀it฀at฀the฀enterprise฀level. 3.฀You฀ not฀ only฀ are฀ expert฀ at฀ number฀ crunching฀ but฀ also฀ invent฀ proprietary฀ metrics฀for฀use฀in฀key฀business฀processes.฀ (p.฀106)

Finding฀employees฀at฀all฀levels฀with฀the฀ necessary฀ quantitative฀ skills฀ is฀ a฀ key฀ problem.

Mathematics฀Proficiency฀in฀the฀ United฀States

We฀have฀not฀found฀statistics฀that฀specifi-cally฀show฀the฀mathematics฀proficiency฀of฀ undergraduate฀ business฀ school฀ students.฀ We฀ must฀ infer฀ this฀ proficiency฀ from฀ the฀ data฀that฀is฀available฀for฀U.S.฀high฀school฀ and฀college฀students฀in฀general.

In฀ 2003,฀ the฀ Organization฀ for฀ Eco-nomic฀ Cooperation฀ and฀ Development’s฀ Program฀ for฀ International฀ Student฀ Assessment฀ performed฀ an฀ internation-al฀ survey฀ of฀ 15-year-olds฀ (Chaddock,฀ 2004).฀ The฀ U.S.฀ 15-year-olds฀ scored฀ measurably฀ better฀ than฀ their฀ counter-parts฀in฀only฀3฀of฀the฀30฀nations฀in฀the฀ Organization฀ for฀ Economic฀

Coopera-tion฀and฀Development.฀Even฀the฀highest฀ U.S.฀ achievers฀ in฀ mathematics฀ literacy฀ and฀problem฀solving฀were฀outperformed฀ by฀ their฀ peers฀ in฀ other฀ industrialized฀ nations.฀

Further,฀once฀in฀college,฀students฀face฀ the฀ following฀ prospect฀ described฀ by฀ a฀ former฀ president฀ of฀ Harvard฀ University:฀ “Most฀ college฀ seniors฀ do฀ not฀ think฀ that฀ they฀ have฀ made฀ substantial฀ progress฀ in฀ improving฀their฀competence฀in฀writing฀or฀ quantitative฀ methods,฀ and฀ some฀ assess-ments฀ have฀ found฀ that฀ many฀ students฀ actually฀regress”฀(Bok,฀2005,฀p.฀1).฀

Quantitative฀Courses฀Required฀at฀ Sample฀U.S.฀Business฀Schools

Prior฀ to฀ suggesting฀ an฀ appropriate฀ curriculum฀for฀quantitative฀literacy,฀it฀is฀ instructive฀to฀examine฀the฀current฀status฀ of฀the฀mathematics฀courses฀required฀of฀ business฀ students฀ at฀ a฀ number฀ of฀ U.S.฀ universities.฀ As฀ we฀ try฀ to฀ decide฀ the฀ minimum฀ acceptable฀ number฀ of฀ hours฀ that฀ each฀ business฀ student฀ should฀ have฀ in฀ mathematics,฀ it฀ is฀ useful฀ to฀ exam-ine฀the฀current฀requirements฀of฀business฀ schools.฀We฀have฀found฀by฀surveying฀a฀ number฀ of฀ business฀ schools฀ that฀ these฀ requirements฀ predominantly฀ include฀ courses฀in฀calculus฀and฀statistics฀of฀3–6฀ semester฀hr฀each.

These฀courses฀do฀not฀normally฀cover฀ some฀ of฀ the฀ essential฀ components฀ of฀ quantitative฀literacy.฀The฀following฀is฀a฀ partial฀list฀of฀quantitative฀literacy฀skills฀ beyond฀arithmetic,฀geometry,฀and฀alge-bra฀ (which฀ are฀ part฀ of฀ every฀ school฀ mathematics฀ program)฀ that฀ the฀ Mathe-matical฀Society฀of฀America฀(Sons,฀1996)฀ endorsed฀and฀that฀we฀believe฀either฀(a)฀ educators฀typically฀do฀not฀include฀in฀the฀ standard฀ calculus฀ and฀ statistics฀ courses฀ or฀ (b)฀ students฀ do฀ not฀ achieve฀ a฀ work-able฀level฀of฀understanding.

1.฀Modeling:฀ Formulating฀problems,฀seek-ing฀patterns,฀and฀drawing฀conclusions;฀ recognizing฀ interactions฀ in฀ complex฀ systems;฀ understanding฀ linear,฀ expo-nential,฀ multivariate,฀ and฀ simulation฀ models;฀ understanding฀ the฀ impact฀ of฀ different฀rates฀of฀growth.

2.฀Chance:฀Recognizing฀that฀seemingly฀฀ improbable฀ coincidences฀ are฀ not฀ uncommon;฀ evaluating฀ risks฀ from฀ available฀evidence;฀understanding฀the฀ value฀of฀random฀samples.

In฀the฀following฀sections,฀we฀elaborate฀on฀ been฀described฀as฀data-drenched.฀Arney฀ (1999)฀argued฀that

The฀ 21st฀century,฀ with฀ the฀ dawning฀ of฀ the฀ information฀ age,฀ brings฀ new฀ tools฀ and฀different฀requirements฀in฀mathemati-cal฀knowledge฀to฀be฀productive.฀Because฀ computers฀can฀be฀used฀to฀shoulder฀much฀ of฀ the฀ computational฀ burden฀ of฀ future฀ work,฀ workers฀ will฀ face฀ a฀ new฀ set฀ of฀ technological฀and฀quantitative฀challenges.฀ (p.฀224)฀

He฀ further฀ stated฀ that฀ understanding฀ complex฀system฀behavior฀is฀one฀of฀the฀ most฀important฀topics฀for฀the฀student฀to฀ learn฀to฀be฀prepared฀for฀the฀complexities฀ of฀the฀21st฀century.฀

The฀problems฀that฀people฀in฀the฀busi-ness฀ world฀ face฀ are฀ complex.฀ To฀ func-tion,฀businesspeople฀create฀a฀simplified฀ representation฀of฀a฀problem฀to฀assist฀in฀ making฀ decisions.฀ This฀ simplified฀ rep-resentation฀ of฀ the฀ problem฀ is฀ a฀ model.฀ A฀ particular฀ type฀ of฀ model฀ of฀ value฀ to฀ business฀ students฀ is฀ a฀ mathematical฀ model,฀ which฀ is฀ an฀ algebraic฀ represen-tation฀ of฀ a฀ situation฀ or฀ problem.฀ The฀ advantage฀ of฀ expressing฀ a฀ problem฀ in฀ algebraic฀terms฀is฀that฀the฀problem฀must฀ be฀explicitly฀defined.฀To฀be฀well฀defined,฀ the฀problem฀must฀be฀well฀understood.฀In฀ fact,฀ one฀ purpose฀ of฀ model฀ building฀ is฀ an฀increased฀understanding฀of฀the฀prob-lem.฀This฀prevents,฀or฀at฀least฀decreases,฀ the฀attempt฀to฀solve฀a฀problem฀without฀ understanding฀ it฀ or฀ trying฀ to฀ solve฀ the฀ wrong฀ problem.฀ See฀ Powell฀ and฀ Baker฀ (2007)฀ for฀ a฀ good฀ introduction฀ to฀ the฀ modeling฀process.

An฀ additional฀ advantage฀ to฀ using฀ mathematical฀models฀to฀represent฀prob-lems฀ is฀ that฀ probmathematical฀models฀to฀represent฀prob-lems฀ of฀ greater฀ com-plexity฀ can฀ be฀ represented฀ and฀ solved.฀ There฀are฀numerous฀classes฀of฀problems฀ that฀ include฀ a฀ large฀ number฀ of฀ deci-sion฀variables฀or฀variables฀with฀a฀large฀ number฀ of฀ possible฀ values.฀ Examples฀ of฀ this฀ type฀ of฀ problem฀ include฀ the฀ many฀ classes฀ of฀ scheduling฀ problems฀ faced฀by฀business฀practitioners,฀includ-ing฀ production฀ scheduling,฀ crew฀ and฀ workforce฀ scheduling,฀ and฀ the฀ routing฀

and฀ scheduling฀ of฀ raw฀ materials฀ and฀ finished฀goods.฀Finding฀good฀solutions฀ to฀such฀problems฀without฀the฀advantage฀ of฀a฀mathematical฀model,฀often฀with฀an฀ associated฀ algorithm,฀ is฀ not฀ practical.฀ See฀ Ragsdale฀ (2007)฀ for฀ a฀ good฀ intro-duction฀to฀a฀number฀of฀these฀models.

฀ In฀ addition,฀ there฀ are฀ problems฀ that฀ are฀complex฀not฀in฀terms฀of฀size฀but฀in฀ terms฀ of฀ complex฀ dynamic฀ behavior.฀ Examples฀ include฀ the฀ behavior฀ of฀ any฀ business฀ system฀ or฀ parts฀ of฀ a฀ business฀ system,฀ including฀ the฀ behavior฀ of฀ sup-ply฀ chains฀ for฀ raw฀ material฀ and฀ fin- ished฀goods,฀for฀the฀manufacturing฀pro-cess฀ and฀ for฀ the฀ supply฀ of฀ labor฀ (e.g.,฀ Manni฀ &฀ Cavana,฀ 2003;฀ McGarvey฀ &฀฀ Hannon,฀ 2004;฀ Pidd,฀ 2004;฀ Sterman,฀ 2000).฀ A฀ mathematical฀ representation฀ of฀ these฀ problems฀ using฀ rate฀ equations฀ and฀ simulation฀ to฀ predict฀ the฀ behavior฀ of฀ the฀ system฀ over฀ time฀ is฀ a฀ way฀ to฀ begin฀to฀understand฀these฀systems.

Opponents฀ of฀ increased฀ quantitative฀ literacy฀ argue฀ that฀ business฀ students฀ do฀ not฀ need฀ mathematical฀ modeling฀ as฀ part฀of฀the฀business฀curriculum฀and฀that฀ modeling฀ is฀ an฀ approach฀ for฀ scientists฀ and฀engineers.฀Contrary฀to฀the฀beliefs฀of฀ this฀group,฀the฀tools฀of฀engineering฀and฀ science฀ are฀ rapidly฀ entering฀ the฀ field฀ of฀ business฀ decision฀ making.฀ A฀ fairly฀ recent฀example฀ is฀ the฀ field฀ of฀ financial฀ engineering.฀ The฀ mathematics฀ used฀ to฀ value฀ options฀ in฀ the฀ field฀ of฀ finance฀ requires฀mathematical฀modeling฀sophis-tication฀ well฀ beyond฀ that฀ acquired฀ by฀ the฀ typical฀ business฀ student฀ in฀ the฀ cur-rent฀curriculum. tal฀ budgeting,฀ cash฀ budgets,฀ risk฀ man- agement,฀workforce฀management,฀ware-house฀location,฀pricing,฀media฀selection,฀ supply฀chain฀analysis฀and฀optimization,฀ and฀so฀on.฀See฀Table฀1฀for฀an฀abbrevi-ated฀ list฀ of฀ functional฀ area฀ problems฀ and฀model฀types฀that฀have฀been฀used฀to฀ guide฀the฀decision-making฀process.

The฀ business฀ world฀ is฀ facing฀ more฀ complicated฀ problems฀ and฀ requires฀ better฀ problem-solving฀ approaches฀ to฀ obtain฀ better฀ solutions.฀ After฀ all฀ busi-ness฀ students’฀ adequate฀ preparation฀ in฀ pure฀ mathematics,฀ the฀ use฀ of฀

math-ematical฀ modeling฀ should฀ be฀ part฀ of฀ their฀ preparation฀ for฀ the฀ 21st฀century.฀ According฀to฀Arney฀(1999),฀they฀will฀be฀ required฀to:

process฀data฀and฀synthesize฀information,฀ use฀ and฀ understand฀ information฀ technol-ogy,฀ optimize฀ elaborate฀ plans,฀ confront฀ complexity,฀ and฀ leverage฀ new฀ technolo-gies.฀ An฀ essential฀ component฀ of฀ mod-ern฀ undergraduate฀ mathematics฀ becomes฀ modeling฀ (formulating฀ and฀ analyzing฀ problems,฀ using฀ technical฀ tools,฀ and฀ implementing฀solutions)฀with฀an฀empha-sis฀ on฀ interdisciplinary฀ problem฀ solving.฀ (p.฀224)฀

Schrage฀ (2000)฀ discussed฀ the฀ impor-tant฀role฀that฀models฀and฀modeling฀play฀ in฀the฀innovation฀process฀of฀companies.฀ The฀ idea฀ is฀ to฀ construct฀ formal฀ models฀ and฀then฀use฀the฀models฀as฀instruments฀ for฀introspection,฀discussion,฀and฀debate.฀ He฀described฀a฀model฀as฀a฀shared฀space฀ that฀ allows฀ this฀ collaboration.฀ In฀ par-ticular,฀ “Any฀ tools,฀ technologies,฀ tech-niques,฀ or฀ toys฀ that฀ let฀ people฀ improve฀ how฀they฀play฀seriously฀with฀uncertainty฀ is฀ guaranteed฀ to฀ improve฀ the฀ quality฀ of฀ innovation”฀ (p.฀ 2).฀ He฀ continued,฀ “how฀ organizations฀ play฀ with฀ their฀ models฀ determines฀how฀successfully฀they฀man-age฀ themselves฀ and฀ their฀ markets”฀ (p.฀ 12).฀ Schrage฀ also฀ pointed฀ out฀ that฀ “the฀ spreadsheet฀transformed฀the฀culture฀and฀ economics฀ of฀ global฀ finance”฀ (p.฀ 12).฀ Last,฀he฀suggested,฀“Whenever฀you฀look฀ for฀ the฀ fundamental฀ dynamics฀ driving฀ innovation,฀you฀find฀innovators฀manag-ing฀models”฀(p.฀12).

Innovation฀ and฀ creativity฀ are฀ essen-tial฀for฀successful฀business฀practice.฀The฀ problem฀ is฀ how฀ to฀ create฀ an฀ environ-ment฀ or฀ a฀ process฀ that฀ will฀ effectively฀ generate฀ creative฀ solutions.฀ These฀ are฀ not฀ created฀ in฀ a฀ vacuum:฀ They฀ usually฀ result฀ from฀ a฀ businessperson’s฀ seeing฀ a฀ problem฀ in฀ a฀ new฀ way฀ or฀ creating฀ a฀ solution฀procedure฀that฀is฀different฀and฀ better.฀What฀role฀do฀models฀and฀model-ing฀play฀in฀this฀creative฀process?

Innovation฀ in฀ any฀ but฀ the฀ simplest฀ of฀ situations฀ can฀ only฀ take฀ place฀ if฀ the฀ problem฀ or฀ process฀ is฀ represented฀ so฀ that฀numerous฀strategies฀or฀options฀can฀ be฀easily฀tried฀and฀evaluated.฀This฀rep-resentation฀ is฀ a฀ model,฀ which฀ is฀ then฀ used฀ as฀ an฀ environment฀ in฀ which฀ to฀ experiment฀with฀alternative฀ideas.฀In฀the฀ business฀ environment,฀ many฀ of฀ these฀฀

representations฀ are฀ quantitative฀ mod-els.฀ A฀ valuable฀ model,฀ in฀ addition฀ to฀ allowing฀ the฀ testing฀ of฀ many฀ alterna-tives,฀ sometimes฀ generates฀ unexpected฀ and฀ surprising฀ results฀ or฀ unanticipat-ed฀ options.฀ For฀ example,฀ consider฀ a฀ company’s฀ supply฀ chain,฀ which฀ needs฀ to฀ be฀ as฀ efficient฀ as฀ possible.฀ There฀ are฀ numerous฀ ways฀ of฀ configuring฀ the฀ chain.฀ Which฀ configuration฀ would฀ be฀ most฀ beneficial?฀Are฀ there฀

unanticipat- ed฀benefits฀from฀a฀particular฀configura- tion?฀No฀one฀can฀explore฀the฀possibili-ties฀without฀a฀quantitative฀model,฀in฀this฀ case฀probably฀a฀stochastic฀simulation.

The฀ point฀ is฀ that฀ innovation฀ cannot฀ take฀ place฀ without฀ the฀ model.฀ Mental฀ models฀ are฀ incomplete,฀ and฀ the฀ for-mal฀ quantitative฀ model฀ is฀ the฀ driver.฀ Consider฀ the฀ relatively฀ unsophisticated฀ spreadsheet.฀Its฀main฀value฀is฀not฀com-putational฀ results฀ per฀ se฀ but฀ the฀ “what฀

if”฀factor:฀the฀ability฀to฀create฀scenarios,฀ explore฀hypothetical฀developments,฀and฀ try฀ out฀ different฀ options.฀ The฀ spread-sheet,฀as฀one฀executive฀said,฀allows฀the฀ users฀to฀create฀and฀then฀experiment฀with฀ “a฀ phantom฀ business฀ within฀ the฀ com-puter”฀ (Schrage,฀ 2000,฀ p.฀ 44).฀ This฀ is฀ how฀the฀quantitative฀model฀makes฀inno-vation฀possible.

Davenport฀(2006)฀described฀the฀wide-spread฀use฀of฀modeling฀and฀optimization฀

TABLE฀1.฀Functional฀Area฀Problems฀and฀Related฀Relevant฀Quantitative฀ Methods

฀ Constrained฀ Risk฀ System

Area฀and฀problems฀ optimizationa_{฀ analysis}b_{฀ dynamics}c

Business

฀ Short-term฀cash฀management฀ _¸฀ _¸฀ ฀ Currency฀trading฀strategies฀ _¸฀ _¸฀

฀ Capital฀budgeting฀ _¸฀ _¸฀

฀ Portfolio฀selection฀ _¸฀ _¸฀฀

฀ Projecting฀cash฀budgets฀ ฀ _¸฀

฀ Retirement฀planning฀ _¸฀ _¸฀

฀ New฀product฀development฀ ฀ _¸฀

฀ Multi-period฀borrowing฀and฀฀

฀ ฀฀฀lending฀ _¸฀ _¸฀ _¸

฀ Managing฀company฀growth฀ ฀ ฀ _¸

฀ Organizational฀structure฀฀

฀ ฀฀฀dynamics฀ ฀ ฀ _¸

Marketing฀ ฀ ฀

฀ Warehouse฀location฀ _¸฀ ฀

฀ Sales฀force฀allocation฀ _¸฀ ฀

฀ Media฀selection฀ _¸฀ ฀

฀ Bidding฀ ฀ _¸฀

฀ Product฀pricing฀ ฀ _¸฀ _¸

฀ Airline฀and฀hotel฀overbooking฀ ฀ _¸฀

฀ Sales฀projection฀ ฀ _¸฀ _¸

฀ Distribution฀strategies฀ _¸฀ _¸฀ ฀ New฀product฀risk฀assessment฀ ฀ _¸฀

฀ Market฀share฀strategy฀ ฀ ฀ _¸

฀ Customer฀interface฀models฀ ฀ ฀ _¸

฀ Managing฀product฀demand฀ ฀ ฀ _¸

฀ Product฀diffusion฀pattern฀ ฀ ฀ _¸

฀ Fad฀and฀fashion฀models฀ ฀ ฀ _¸

฀ Product฀life฀cycle฀models฀ ฀ ฀ _¸

Operation฀and฀supply฀chain฀ ฀ ฀

฀ Product฀mix฀ _¸฀ ฀

฀ Product฀scheduling฀ _¸฀ ฀

฀ Production฀planning฀ _¸฀ ฀

฀ Machine฀scheduling฀ _¸฀ ฀

฀ Facility฀location฀ _¸฀ ฀

฀ Project฀management฀ _¸฀ _¸฀

฀ Center฀capacity฀analysis฀ ฀ _¸฀ _¸

฀ System฀configuration฀ ฀ ฀ _¸

฀ Supplier฀interface฀models฀ ฀ ฀ _¸

฀ Supply฀chain฀models฀ _¸฀ _¸฀ _¸

Note.฀Sources฀for฀functional฀area฀examples฀are฀F.฀W.฀Winston฀and฀S.฀C.฀Albright฀(1997),฀J.฀Evans฀ and฀D.฀Olson฀(2002),฀B.฀McGarvey฀and฀B.฀Hannon฀(2004),฀and฀J.฀D.฀Sterman฀(2000).

a

Includes฀linear฀programming,฀integer฀programming,฀nonlinear฀programming,฀and฀network฀mod-els.฀b_{Includes฀decision฀trees,฀Monte฀Carlo฀simulation,฀and฀queuing฀simulation.฀}c_{Includes฀discrete฀}

system฀analytical฀methods฀and฀system฀simulation฀methods.

in฀ the฀ companies฀ that฀ he฀ studied.฀ He฀ gave฀several฀examples:฀predictive฀mod- eling฀to฀identify฀the฀most฀profitable฀cus-tomers฀ plus฀ those฀ with฀ the฀ most฀ profit฀ potential,฀optimization฀of฀supply฀chains,฀ and฀establishment฀of฀prices฀in฀real฀time฀ to฀ get฀ the฀ highest฀ yield฀ possible฀ from฀ each฀customer฀transaction.

In฀ essence,฀ the฀ student฀ of฀ today฀ requires฀ a฀ curriculum฀ that฀ does฀ not฀ focus฀ on฀ computational฀ methods฀ of฀ mathematics฀ but฀ on฀ problem-solving฀ methods฀and฀the฀use฀of฀mathematics฀as฀ unknown฀ and฀ uncertain.฀ Risk฀ manage-ment,฀ which฀ assumes฀ that฀ future฀ risks฀ can฀ be฀ understood,฀ measured,฀ and—to฀ some฀ extent—predicted,฀ is฀ at฀ the฀ core฀ of฀ fields฀ as฀ diverse฀ as฀ business฀ fore-casting,฀ portfolio฀ theory,฀ odds฀ making,฀ insurance฀ and฀ derivatives,฀ new฀ product฀ development,฀ capital฀ investment,฀ mar-ket฀ development,฀ and฀ global฀ expansion.฀ Bernstein฀(1998)฀indicated,฀“The฀essence฀ of฀ risk฀ management฀ lies฀ in฀ maximiz- ing฀the฀areas฀where฀we฀have฀some฀con-trol฀over฀the฀outcome฀while฀minimizing฀ the฀ areas฀ where฀ we฀ have฀ absolutely฀ no฀ control฀ over฀ the฀ outcome฀ and฀ the฀ link-age฀ between฀ effort฀ and฀ cause฀ is฀ hidden฀ from฀ us”฀ (p.฀ 107).฀Control฀ is฀ the฀ result฀ of฀a฀knowledge฀or฀understanding฀of฀the฀ cause฀and฀effect฀relations฀that฀are฀inher-ent฀ in฀ the฀ structure฀ of฀ the฀ problem฀ or฀ situation.฀People฀have฀no฀control฀in฀some฀ parts฀of฀the฀problem฀because฀they฀do฀not฀ have฀that฀understanding.฀Businesspeople฀ typically฀ characterize฀ such฀ parts฀ of฀ the฀ problem฀as฀uncertain฀and฀try฀to฀quantify฀ that฀uncertainty฀by฀the฀use฀of฀probabili-ties.฀ The฀ business฀ decision฀ maker฀ then฀ has฀ the฀ task฀ of฀ making฀ decisions฀ under฀ the฀ conditions฀ just฀ described.฀ The฀ use฀ of฀ the฀ appropriate฀ methods฀ and฀ mod-els฀ available฀ for฀ decision฀ making฀ under฀ these฀conditions฀can฀greatly฀improve฀the฀ decision-making฀ process.฀ Frequently,฀ a฀ model฀ in฀ conjunction฀ with฀ computer฀ simulation฀ is฀ used฀ as฀ a฀ means฀ toward฀ better฀ analysis฀ and฀ decision฀ making฀ for฀ these฀types฀of฀problems.

A฀Proposal฀to฀Meet฀the฀ Quantitative฀Literacy฀Needs฀ of฀Business฀Students

Because฀ of฀ the฀ aforementioned฀ need฀ for฀additional฀quantitative฀tools฀for฀busi-ness฀students฀to฀be฀adequately฀prepared฀ for฀ the฀ future,฀ the฀ question฀ about฀ how฀ this฀ can฀ be฀ achieved฀ remains.฀ Students฀ ultimately฀need฀to฀be฀prepared฀to฀solve฀ practical฀ problems฀ by฀ applying฀ math-ematical฀ concepts฀ that฀ are฀ relevant.฀As฀ discussed฀ in฀ the฀ previous฀ section,฀ this฀ requirement฀indicates฀a฀need฀for฀them฀to฀ be฀able฀to฀construct฀and฀use฀models฀for฀ solving฀business฀problems.฀They฀should฀ also฀be฀prepared฀to฀respond฀to฀complex฀ system฀ behavior,฀ which฀ accompanies฀ most฀business฀situations.฀An฀introduction฀ to฀optimization฀as฀part฀of฀the฀instruction฀ in฀model฀building฀is฀warranted฀because฀ businesspeople฀are฀trying฀to฀find฀the฀best฀ solutions฀ to฀ problems.฀ Last,฀ a฀ student฀ should฀ be฀ introduced฀ to฀ working฀ with฀ uncertainty฀and฀how฀to฀make฀good฀deci-sions฀even฀if฀they฀are฀uncertain.฀

The฀ calculus฀ course฀ provides฀ the฀ fun-damental฀ mathematical฀ underpinning฀ of฀ rates฀ of฀ change฀ and฀ accumulation฀ nec-essary฀ for฀ a฀ student฀ to฀ begin฀ to฀ model฀ the฀ behavior฀ of฀ complex฀ systems.฀ It฀ is฀ imperative฀ that฀ this฀ course฀ be฀ presented฀ so฀ that฀ the฀ student฀ sees฀ the฀ connection฀ between฀the฀use฀of฀calculus฀and฀the฀solv-ing฀of฀business฀problems.฀The฀bridge,฀in฀ our฀opinion,฀is฀to฀include฀modeling฀as฀part฀ of,฀ or฀ in฀ conjunction฀ with,฀ the฀ calculus฀ course.฀ The฀ discussion฀ would฀ focus฀ on฀ building฀ simple฀ models฀ that฀ involve฀ rate฀ equations.฀A฀ simple฀ example฀ of฀ the฀ use฀ of฀rate฀equations฀in฀business฀is฀estimating฀ a฀ system฀ over฀ time.฀ This฀ discussion฀ can฀ be฀the฀link฀showing฀the฀value฀of฀calculus฀ for฀ problem฀ solving.฀We฀ do฀ not฀ propose฀ that฀ much฀ time฀ be฀ spent฀ on฀ analytical฀ methods฀for฀solving฀these฀models฀beyond฀ some฀very฀simple฀ones.฀Computer฀algebra฀ software฀ or฀ simulation฀ methods,฀ or฀ even฀ spreadsheets,฀can฀be฀used฀for฀this฀purpose.฀ For฀ other฀ examples฀ of฀ such฀ models,฀ see฀ Giordano,฀Weir,฀and฀Fox฀(2003).฀

We฀ believe฀ that฀ educators฀ and฀ stu-dents฀ can฀ cover฀ most฀ of฀ this฀ material,฀ including฀the฀calculus,฀in฀about฀6฀semes-ably฀ the฀ most฀ creative฀ way฀ to฀ accom-plish฀ this,฀ but฀ for฀ many฀ institutions฀ this฀ approach฀may฀not฀be฀workable.฀Instead,฀ a฀ practical฀ approach฀ is฀ simply฀ to฀ add฀ a฀ required฀ modeling฀ course฀ to฀ the฀ cur-riculum฀for฀all฀students.฀The฀course฀must฀ focus฀on฀using฀modeling฀to฀solve฀relevant฀ functional฀area฀problems.฀In฀addition,฀the฀ course฀should฀be฀the฀bridge฀that฀ties฀the฀ preparation฀ in฀ calculus฀ to฀ the฀ solving฀ of฀ business฀ problems.฀ The฀ use฀ of฀ the฀ spreadsheet฀ as฀ a฀ modeling฀ environment฀ would฀ certainly฀ improve฀the฀chances฀of฀ seeing฀increased฀use฀of฀modeling฀in฀the฀ functional฀ areas.฀ Thus,฀ the฀ ideal฀ course฀ would฀focus฀on฀business฀problems฀with฀ the฀use฀of฀modeling฀demonstrated฀as฀the฀ route฀to฀better฀decisions.฀

Ideally,฀the฀students฀should฀see฀mod-eling฀ across฀ the฀ curriculum,฀ which฀ means฀the฀use฀of฀modeling฀and฀models฀ in฀the฀functional฀area฀courses฀as฀well.฀A฀ bridge฀must฀be฀built฀between฀the฀quan-titative฀ and฀ functional฀ areas฀ to฀ allow฀ this฀to฀happen.฀The฀functional฀area฀fac-ulty,฀including฀the฀administration,฀must฀ be฀ convinced฀ that฀ quantitative฀ literacy฀ is฀ invaluable฀ in฀ achieving฀ better฀ busi-ness฀decisions.฀The฀work฀of฀Davenport฀ (2006)฀and฀others฀must฀be฀used฀as฀sales฀ tools,฀ along฀ with฀ data฀ about฀ trends฀ in฀ industry,฀ to฀ convince฀ others฀ that฀ the฀ work฀is฀important.

Although฀ this฀ process฀ seems฀ difficult฀ and฀ requires฀ much฀ commitment฀ and฀ effort,฀ we฀ believe฀ the฀ results฀ could฀ be฀ impressive.฀The฀ objective฀ of฀ integrating฀ modeling฀ into฀ the฀ curriculum฀ and฀ the฀ process฀that฀we฀have฀suggested฀reflect฀the฀ plans฀of฀a฀number฀of฀business฀schools฀to฀ integrate฀the฀functional฀areas฀of฀business.฀ The฀ proposed฀ procedure฀ for฀ improving฀ quantitative฀ literacy฀ could฀ easily฀ piggy-back฀on฀the฀overall฀plan฀for฀integration.

A฀ widely฀ acclaimed฀ integrative฀ approach฀that฀includes฀many฀of฀the฀ele-ments฀that฀we฀believe฀are฀important฀for฀

business฀students฀as฀part฀of฀the฀attempt฀to฀ improve฀quantitative฀literacy฀was฀devel- oped฀over฀the฀last฀few฀years฀at฀the฀Uni-versity฀of฀Arizona฀under฀the฀auspices฀of฀ a฀ multiyear฀ program฀ sponsored฀ by฀ the฀ Mathematical฀Association฀of฀America.฀It฀ in฀ part฀ two.฀ Six฀ other฀ institutions฀ had฀ used฀their฀material฀as฀of฀November฀2002฀ (Albers,฀ 2002).฀ In฀ addition,฀ a฀ number฀ of฀business฀schools฀teach฀modeling฀in฀a฀ stand-alone฀ required฀ or฀ elective฀ course฀ that฀may฀be฀titled฀Management฀Science฀ or฀Operations฀Research.

Conclusion

In฀ trying฀ to฀ promote฀ the฀ importance฀ of฀quantitative฀literacy฀for฀business฀stu-dents,฀we฀appreciate฀that฀we฀are฀in฀some฀ sense฀ trying฀ to฀ change฀ a฀ culture฀ that฀ believes฀that฀mathematics฀is฀not฀valuable฀ for฀ business฀ students.฀ The฀ problem฀ is฀ more฀widespread฀than฀in฀just฀the฀business฀ community:฀It฀is฀ingrained฀in฀the฀popula-tion฀ at฀ large.฀A฀ number฀ of฀ years฀ ago,฀ a฀ president฀of฀the฀American฀Mathematical฀ Association฀ pointed฀ out฀ that฀ people฀ are฀ ashamed฀ of฀ being฀ verbally฀ illiterate฀ but฀ do฀not฀seem฀to฀possess฀the฀same฀level฀of฀ guilt฀for฀being฀mathematically฀illiterate.฀ In฀fact,฀many฀brag฀about฀it.฀

In฀ the฀ present฀ study,฀ we฀ found฀ the฀ following:

1.฀The฀ United฀ States฀ is฀ behind฀ the฀ rest฀ of฀ the฀ industrialized฀ world฀ in฀ terms฀ of฀quantitative฀literacy.

2.฀This฀ circumstance฀ is฀ true฀ not฀ only฀ for฀ the฀ average฀ student฀ but฀ also฀ for฀ students฀admitted฀to฀selective฀univer-sities฀and฀their฀business฀schools. 3.฀Quantitative฀ methods฀ courses฀ have฀

not฀ changed฀ much฀ in฀ half฀ a฀ century,฀ although฀ the฀ business฀ environment฀

has฀evolved฀dramatically฀with฀devel-5.฀High฀ school฀ mathematics฀ followed฀ by฀ college฀ courses฀ in฀ calculus฀ and฀ statistics฀are฀insufficient฀for฀quantita-tive฀literacy.

6.฀Modeling฀ and฀ risk฀ management฀ are฀ vital฀ aspects฀ of฀ quantitative฀ literacy฀ that฀are฀missed฀by฀focusing฀solely฀on฀ calculus฀and฀statistics.

7.฀Heavy฀use฀should฀be฀made฀of฀widely฀ available฀ computer฀ software฀ in฀ busi-ness฀ schools฀ to฀ more฀ easily฀ apply฀ quantitative฀ methods฀ to฀ business฀ problems฀ and฀ to฀ apply฀ sophisticated฀ analyses฀to฀large฀data฀sets.

In฀conclusion,฀it฀is฀important฀for฀edu-cators฀ to฀ remember฀ that฀ “for฀ most฀ stu-dents,฀skills฀learned฀free฀of฀context฀are฀ skills฀devoid฀of฀meaning฀and฀utility.฀To฀ be฀ effective,฀ numeracy฀ skills฀ must฀ be฀ taught฀ and฀ learned฀ in฀ settings฀ that฀ are฀ both฀ meaningful฀ and฀ memorable”฀ (QL฀ Design฀Team,฀2001).฀

NOTES

Richard฀McClure ฀is฀professor฀of฀decision฀sci-ences฀in฀the฀Farmer฀School฀of฀Business฀at฀Miami฀ University.

Sumit฀ Sircar฀ is฀ the฀ Armstrong฀ Professor฀ of฀ communications฀ technology฀ and฀ management฀ in฀ the฀ Farmer฀ School฀ of฀ Business฀ at฀ Miami฀ Uni-versity.

Correspondence฀ concerning฀ this฀ article฀ should฀ be฀ addressed฀ to฀ Dr.฀ Richard฀ McClure,฀ Professor฀ of฀Decision฀Sciences,฀Farmer฀School฀of฀Business,฀ Miami฀University,฀Oxford,฀OH฀45056,฀USA.฀

E-mail:฀mcclurrh@muohio.edu฀

REFERENCES

Albers,฀D.฀J.฀(2002,฀November).฀A฀genuine฀inter-disciplinary฀ partnership:฀ MAA฀ unveils฀ math-ematics฀ for฀ business฀ decisions.฀Focus,฀ Math-ematical฀Association฀of฀America,฀15.฀ Arney,฀D.฀C.฀(1999).฀Undergraduate฀mathematics฀

for฀the฀future:฀Modeling฀and฀solving฀problems,฀ understanding฀ the฀ new฀ sciences.฀ Retrieved฀ March฀ 20,฀ 2006,฀ from฀ http://www.projectinter฀ math.org/docs/arney1.pdf

Baker,฀S.฀(2006,฀January฀23).฀Why฀math฀will฀rock฀ your฀world.฀Business฀Week,฀54–62.

Bernstein,฀ P.฀ (1998).฀Against฀ the฀ gods:฀ The฀ remarkable฀story฀of฀risk.฀New฀York:฀Bernstein.฀ Bok,฀ D.฀ (2005).฀Are฀ colleges฀ failing?฀ Higher฀ ed฀ needs฀new฀lesson฀plans.฀Retrieved฀December฀18,฀ 2005,฀from฀http://www.boston.com/news/globe/ editorial_opinion/oped/articles/2005/12/18/ Chaddock,฀ G.฀ R.฀ (2004).฀Math฀ +฀ test฀ =฀Trouble฀

for฀US฀economy.฀Retrieved฀December฀7,฀2004,฀ from฀ http://www.csmonitor.com/2004/1207/ p01s04-ussc.html

Davenport,฀ T.฀ H.฀ (2006,฀ January).฀ Competing฀ on฀ analytics.฀Harvard฀ Business฀ Review,฀84,฀ 98–107.

Davenport,฀T.฀H.,฀&฀Harris,฀J.฀G.฀(2007).฀ Compet-ing฀ on฀ analytics:฀The฀ new฀ science฀ of฀ winning.฀ Boston:฀Harvard฀Business฀School.฀

Dingwall,฀J.฀(2000).฀International฀Life฀Skills฀Sur-vey.฀Policy฀research฀initiative.฀Ottawa,฀Ontario,฀ Canada:฀Statistics฀Canada.

Evans,฀ J.,฀ &฀ Olson,฀ D.฀ (2002).฀Simulation฀ and฀ risk฀analysis฀(2nd฀ed.).฀Upper฀Saddle฀River,฀NJ:฀ Prentice฀Hall.฀

Giordano,฀F.฀R.,฀Weir,฀M.฀D.,฀&฀Fox,฀W.฀P.฀(2003).฀ A฀ first฀ course฀ in฀ mathematical฀ modeling฀(3rd฀ ed.).฀Pacific฀Grove,฀CA:฀Brooks/Cole.฀ Kolata,฀ G.฀ (1997).฀ Understanding฀ the฀ news.฀ In฀

L.฀ A.฀ Steen฀ (Ed.),฀Quantitative฀ literacy฀ for฀ tomorrow’s฀ America฀(pp.฀ 23–29).฀New฀ York:฀ The฀College฀Board.฀

Lange,฀ J.฀ D.฀ (2003).฀ Mathematics฀ for฀ literacy.฀ In฀ B.฀ L.฀ Madison฀ &฀ L.฀ A.฀ Steen฀ (Eds.),฀ Quantitative฀ literacy฀(p.฀ 81).฀ Princeton,฀ NJ:฀ National฀ Council฀ on฀ Education฀ and฀ the฀ Dis-ciplines.

Manni,฀ K.฀ E.,฀ &฀ Cavana,฀ R.฀Y.฀ (2003).฀Systems฀ thinking฀and฀modeling:฀Understanding฀change฀ and฀ complexity.฀Albany,฀ Auckland,฀ New฀ Zea-land:฀Pearson฀Prentice฀Hall.฀

McGarvey,฀ B.,฀ &฀ Hannon,฀ B.฀ (2004).฀Dynamic฀ modeling฀ for฀ business฀ management:฀ An฀ intro-duction.฀New฀York:฀Springer.฀

Pidd,฀M.฀(Ed.).฀(2004).฀Systems฀modeling฀theory฀ and฀practice.฀West฀Sussex,฀England:฀Wiley.฀ Powell,฀ S.฀ G.,฀ &฀ Baker,฀ K.฀ R.฀ (2007).฀

Manage-ment฀science:฀ The฀art฀of฀modeling฀with฀spread-sheets฀(2nd฀ed.).฀Hoboken,฀NJ:฀Wiley.

QL฀Design฀Team.฀(2001).฀The฀case฀for฀quantita-tive฀literacy.฀In฀L.฀A.฀Steen฀(Ed.),฀Mathematics฀ and฀democracy฀(p.฀16).฀Princeton,฀NJ:฀National฀ Council฀on฀Education฀and฀the฀Disciplines.฀ Ragsdale,฀C.฀T.฀(2007).฀Spreadsheet฀modeling฀and฀

decision฀analysis฀(5th฀ed.). ฀Mason,฀OH:฀Thom-son฀South-Western.฀

Richardson,฀ R.฀ M.,฀ &฀ McCallum,฀W.฀ G.฀ (2004).฀ In฀ L.฀ A.฀ Steen฀ (Ed.),฀ Achieving฀ quantitative฀ literacy.฀Mathematical฀Association฀of฀America,฀ MAA฀Notes฀[report]฀62,฀14.

Schrage,฀ M.฀ (2000).฀Serious฀ play:฀ How฀ the฀ world’s฀ best฀ companies฀ simulate฀ to฀ innovate.฀ Boston:฀Harvard฀Business฀School฀Press. Sons,฀ L.,฀ Ed.฀ (1996).฀Quantitative฀ reasoning฀

for฀ college฀ graduates:฀ A฀ supplement฀ to฀ the฀ standards.฀MAA฀ Report฀ 1,฀ A฀ report฀ of฀ the฀ CUPM฀ Committee฀ on฀ Literacy฀ Requirements.฀ Washington,฀DC:฀Mathematical฀Association฀of฀ America.

Steen,฀ L.฀ A.฀ (2004).฀ Achieving฀ quantitative฀ lit-eracy.฀Mathematical฀ Association฀ of฀ America,฀ MAA฀Notes฀[report],฀62,฀xii.฀

Sterman,฀J.฀D.฀(2000).฀Business฀dynamics,฀systems฀ thinking฀ and฀ modeling฀ for฀ a฀ complex฀ world.฀ Boston:฀McGraw-Hill.

Winston,฀ F.฀ W.,฀ &฀ Albright,฀ S.฀ C.฀ (1997).฀ Practical฀ management฀ science,฀ spreadsheet฀ modeling฀ and฀ applications.฀ Belmont,฀ CA:฀ Duxbury.