Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji joeb.83.6.369-374

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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Downloaded by [Universitas Maritim Raja Ali Haji] at 23:16 11 January 2016

Quantitative฀Literacy฀for฀Undergraduate฀
Business฀Students฀in฀the฀21st฀Century
RICHARD฀McCLURE
SUMIT฀SIRCAR
MIAMI฀UNIVERSITY
OXFORD,฀OHIO฀

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฀competitive฀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฀propose฀that฀educators฀prepare฀students฀in฀the฀
areas฀of฀mathematical฀modeling฀and฀risk฀
management฀and฀quantitative฀skills,฀teaching฀them฀in฀the฀context฀of฀meaningful฀business฀problems.
Keywords:฀business฀students,฀mathematical฀
modeling,฀quantitative฀literacy
Copyright฀©฀2008฀Heldref฀Publications

฀

T

he฀ environment฀ in฀ which฀ business฀ 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฀situation฀in฀which฀companies฀generally฀produce฀ similar฀ products฀ and฀ have฀ access฀
to฀similar฀technologies.฀Therefore,฀companies฀ must฀ compete฀ by฀ differentiating฀
their฀ business฀ processes,฀ requiring฀ the฀
widespread฀use฀of฀business฀analytics฀for฀
effectiveness฀(Davenport,฀2006;฀Davenport฀&฀Harris,฀2007).
The฀central฀theme฀of฀this฀article฀is฀that฀
quantitative฀ methods฀ can฀ and฀ should฀ be฀
applied฀to฀a฀wide฀array฀of฀decision-making฀scenarios฀and฀that฀all฀business฀students฀
should฀have฀an฀adequate฀level฀of฀quantitative฀literacy฀to฀make฀calculated฀decisions฀
in฀the฀increasingly฀complex,฀informationoriented,฀ knowledge-based฀ world.฀ We฀
subscribe฀ to฀ the฀ definition฀ of฀ quantitative฀literacy฀adopted฀by฀the฀International฀
Life฀Skills฀Survey฀(Dingwall,฀2000):฀“An฀

aggregate฀ of฀ skills,฀ knowledge,฀ beliefs,฀
dispositions,฀ habits฀ of฀ mind,฀ communication฀ capabilities,฀ and฀ problem฀ solving฀
skills฀that฀people฀need฀in฀order฀to฀engage฀
effectively฀ in฀ quantitative฀ situations฀ arising฀in฀life฀and฀work”฀(p.฀147).฀
Although฀ the฀ term฀ quantitative฀ literacy฀ is฀ a฀ superset฀ of฀ the฀ term฀ numeracy฀

(Lange,฀2003),฀we฀use฀them฀interchangeably.฀ We฀ strongly฀ believe฀ that฀ numeracy฀ relates฀ to฀ numbers฀ exactly฀ as฀ literacy฀relates฀to฀words.฀College฀education฀
should฀stress฀the฀two฀equally,฀but฀such฀an฀
equal฀stress฀does฀not฀occur฀at฀most฀institutions.฀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฀ mathematics฀ department฀ any฀ more฀ than฀ educators฀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.฀ competitiveness฀(only฀6฀of฀the฀top฀25฀information฀ technology฀ companies฀ are฀ based฀
in฀the฀United฀States);฀globalization฀and฀
outsourcing฀ to฀ foreign฀ countries;฀ the฀
threat฀of฀India,฀China,฀and฀South฀Korea฀
July/August฀2008฀

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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฀economy฀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฀ literacy฀ of฀ business฀ students฀ because฀ they฀
believe฀that฀(a)฀all฀business฀students฀do฀
not฀need฀much฀mathematics฀beyond฀the฀
high฀school฀level฀except฀for฀a฀course฀in฀
statistics฀and฀(b)฀calculus฀is฀an฀unnecess-฀
ary฀hurdle.฀“I’ve฀never฀used฀calculus฀in฀
all฀these฀years”฀is฀a฀common฀refrain฀that฀
we฀have฀heard฀from฀faculty.฀
This฀reluctance฀to฀increase฀the฀emphasis฀on฀quantitative฀literacy฀has฀resulted฀in฀
it฀ being฀ practically฀ nonexistent฀ in฀ business฀curricula฀(Kolata,฀1997).฀Our฀objective฀ for฀ this฀ article฀ is฀ to฀ argue฀ that฀ to฀
compete฀globally฀and฀prepare฀American฀
business฀students฀for฀the฀future,฀the฀following฀ 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฀sections,฀ we฀ describe฀ the฀ emerging฀ impact฀
of฀quantitative฀methods฀in฀business,฀highlight฀ 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฀undergraduate฀ 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฀engineering,฀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฀
370฀

Journal฀of฀Education฀for฀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฀ graphics฀ 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,฀factbased฀analysis,฀Davenport฀(2006)฀found฀
that฀ virtually฀ all฀ were฀ leaders฀ in฀ their฀
fields.฀ They฀ emphasized฀ business฀ analytics฀as฀an฀overarching฀strategy฀championed฀by฀their฀top฀leadership,฀and฀those฀
organizations฀pushed฀this฀strategy฀down฀
to฀decision฀making฀at฀every฀level.฀Three฀
of฀his฀recommendations฀are฀particularly฀
relevant฀to฀the฀present฀article:
1.฀You฀hire฀not฀only฀people฀with฀analytical฀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฀specifically฀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฀ Economic฀ Cooperation฀ and฀ Development’s฀
Program฀ for฀ International฀ Student฀
Assessment฀ performed฀ an฀ international฀ survey฀ of฀ 15-year-olds฀ (Chaddock,฀
2004).฀ The฀ U.S.฀ 15-year-olds฀ scored฀
measurably฀ better฀ than฀ their฀ counterparts฀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฀ assessments฀ 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฀ examine฀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฀algebra฀ (which฀ are฀ part฀ of฀ every฀ school฀
mathematics฀ program)฀ that฀ the฀ Mathematical฀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฀ workable฀level฀of฀understanding.

1.฀Modeling:฀Formulating฀problems,฀seeking฀patterns,฀and฀drawing฀conclusions;฀
recognizing฀ interactions฀ in฀ complex฀
systems;฀ understanding฀ linear,฀ exponential,฀ 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฀
the฀importance฀of฀modeling฀and฀risk฀management฀and฀the฀issues฀that฀they฀cover.
The฀Need฀for฀Modeling฀in฀the฀
Business฀Curriculum

Downloaded by [Universitas Maritim Raja Ali Haji] at 23:16 11 January 2016

The฀current฀business฀environment฀has฀
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฀mathematical฀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฀business฀ world฀ face฀ are฀ complex.฀ To฀ function,฀businesspeople฀create฀a฀simplified฀
representation฀of฀a฀problem฀to฀assist฀in฀
making฀ decisions.฀ This฀ simplified฀ representation฀ of฀ the฀ problem฀ is฀ a฀ model.฀
A฀ particular฀ type฀ of฀ model฀ of฀ value฀
to฀ business฀ students฀ is฀ a฀ mathematical฀
model,฀ which฀ is฀ an฀ algebraic฀ representation฀ 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฀problem.฀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฀problems฀ is฀ that฀ problems฀ of฀ greater฀ complexity฀ can฀ be฀ represented฀ and฀ solved.฀
There฀are฀numerous฀classes฀of฀problems฀
that฀ include฀ a฀ large฀ number฀ of฀ decision฀ 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,฀including฀ 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฀ introduction฀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฀ supply฀ chains฀ for฀ raw฀ material฀ and฀ finished฀goods,฀for฀the฀manufacturing฀process฀ 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฀sophistication฀ well฀ beyond฀ that฀ acquired฀ by฀
the฀ typical฀ business฀ student฀ in฀ the฀ current฀curriculum.
Mathematical฀models฀are฀applied฀frequently฀in฀many฀of฀the฀more฀sophisticated฀business฀organizations.฀The฀types฀of฀
problems฀that฀are฀attacked฀using฀models฀
include฀business฀activities,฀such฀as฀capital฀ budgeting,฀ cash฀ budgets,฀ risk฀ management,฀workforce฀management,฀warehouse฀location,฀pricing,฀media฀selection,฀
supply฀chain฀analysis฀and฀optimization,฀
and฀ so฀ on.฀ See฀Table฀ 1฀ for฀ an฀ abbreviated฀ 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฀ business฀ 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฀ technology,฀ optimize฀ elaborate฀ plans,฀ confront฀
complexity,฀ and฀ leverage฀ new฀ technologies.฀ An฀ essential฀ component฀ of฀ modern฀ undergraduate฀ mathematics฀ becomes฀
modeling฀ (formulating฀ and฀ analyzing฀
problems,฀ using฀ technical฀ tools,฀ and฀
implementing฀ solutions)฀ with฀ an฀ emphasis฀ on฀ interdisciplinary฀ problem฀ solving.฀
(p.฀224)฀

Schrage฀ (2000)฀ discussed฀ the฀ important฀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฀ particular,฀ “Any฀ tools,฀ technologies,฀ techniques,฀ 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฀ manage฀ 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฀ managing฀models”฀(p.฀12).
Innovation฀ and฀ creativity฀ are฀ essential฀for฀successful฀business฀practice.฀The฀
problem฀ is฀ how฀ to฀ create฀ an฀ environment฀ 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฀modeling฀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฀representation฀ is฀ a฀ model,฀ which฀ is฀ then฀
used฀ as฀ an฀ environment฀ in฀ which฀ to฀
experiment฀with฀alternative฀ideas.฀In฀the฀
business฀ environment,฀ many฀ of฀ these฀฀
July/August฀2008฀

371

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

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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฀

Constrained฀
optimizationa฀

Risk฀
analysisb฀

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System
dynamicsc

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¸

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฀models.฀bIncludes฀decision฀trees,฀Monte฀Carlo฀simulation,฀and฀queuing฀simulation.฀cIncludes฀discrete฀
system฀analytical฀methods฀and฀system฀simulation฀methods.

representations฀ are฀ quantitative฀ models.฀ A฀ valuable฀ model,฀ in฀ addition฀ to฀
allowing฀ the฀ testing฀ of฀ many฀ alternatives,฀ sometimes฀ generates฀ unexpected฀
and฀ surprising฀ results฀ or฀ unanticipated฀ 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฀ unanticipat372฀

Journal฀of฀Education฀for฀Business

ed฀benefits฀from฀a฀particular฀configuration?฀No฀one฀can฀explore฀the฀possibilities฀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฀ formal฀ quantitative฀ model฀ is฀ the฀ driver.฀
Consider฀ the฀ relatively฀ unsophisticated฀
spreadsheet.฀Its฀main฀value฀is฀not฀computational฀ results฀ per฀ se฀ but฀ the฀ “what฀

if”฀factor:฀the฀ability฀to฀create฀scenarios,฀
explore฀hypothetical฀developments,฀and฀
try฀ out฀ different฀ options.฀ The฀ spreadsheet,฀as฀one฀executive฀said,฀allows฀the฀
users฀to฀create฀and฀then฀experiment฀with฀
“a฀ phantom฀ business฀ within฀ the฀ computer”฀ (Schrage,฀ 2000,฀ p.฀ 44).฀ This฀ is฀
how฀the฀quantitative฀model฀makes฀innovation฀possible.
Davenport฀(2006)฀described฀the฀widespread฀use฀of฀modeling฀and฀optimization฀

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in฀ the฀ companies฀ that฀ he฀ studied.฀ He฀
gave฀ several฀ examples:฀ predictive฀ modeling฀to฀identify฀the฀most฀profitable฀customers฀ 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฀
a฀problem-solving฀tool.
The฀Need฀to฀Cover฀Chance฀
and฀Risk฀Management฀in฀the฀
Business฀Curriculum
Almost฀all฀business฀decisions฀embody฀
an฀element฀of฀risk฀because฀the฀future฀is฀
unknown฀ and฀ uncertain.฀ Risk฀ management,฀ which฀ assumes฀ that฀ future฀ risks฀
can฀ be฀ understood,฀ measured,฀ and—to฀
some฀ extent—predicted,฀ is฀ at฀ the฀ core฀
of฀ fields฀ as฀ diverse฀ as฀ business฀ forecasting,฀ portfolio฀ theory,฀ odds฀ making,฀
insurance฀ and฀ derivatives,฀ new฀ product฀
development,฀ capital฀ investment,฀ market฀ development,฀ and฀ global฀ expansion.฀
Bernstein฀(1998)฀indicated,฀“The฀essence฀
of฀ risk฀ management฀ lies฀ in฀ maximizing฀the฀areas฀where฀we฀have฀some฀control฀over฀the฀outcome฀while฀minimizing฀
the฀ areas฀ where฀ we฀ have฀ absolutely฀ no฀
control฀ over฀ the฀ outcome฀ and฀ the฀ linkage฀ 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฀inherent฀ 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฀probabilities.฀ The฀ business฀ decision฀ maker฀ then฀
has฀ the฀ task฀ of฀ making฀ decisions฀ under฀
the฀ conditions฀ just฀ described.฀ The฀ use฀
of฀ the฀ appropriate฀ methods฀ and฀ models฀ 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฀business฀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฀ mathematical฀ 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฀decisions฀even฀if฀they฀are฀uncertain.฀
The฀ calculus฀ course฀ provides฀ the฀ fundamental฀ mathematical฀ underpinning฀ of฀
rates฀ of฀ change฀ and฀ accumulation฀ necessary฀ 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฀solving฀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฀
the฀growth฀of฀principal฀over฀time฀by฀using฀
continuous฀compounding.฀The฀same฀simple฀type฀of฀model฀can฀be฀used฀to฀represent฀
the฀growth฀of฀other฀phenomena฀over฀time,฀
such฀ as฀ demand฀ for฀ a฀ product฀ or฀ growth฀
of฀a฀population.฀These฀rate฀equations฀are฀
the฀models฀that฀represent฀the฀behavior฀of฀
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฀ students฀ can฀ cover฀ most฀ of฀ this฀ material,฀
including฀the฀calculus,฀in฀about฀6฀semester฀hr,฀in฀addition฀to฀preparation฀in฀business฀statistics.฀Also,฀it฀is฀important฀that฀
business฀courses฀in฀the฀functional฀areas฀
begin฀to฀use฀these฀methods฀as฀part฀of฀the฀
business฀problem-solving฀process.
What฀will฀it฀take฀to฀improve฀the฀quantitative฀literacy฀of฀business฀students?฀The฀
integrative฀approach฀we฀describe฀is฀probably฀ the฀ most฀ creative฀ way฀ to฀ accomplish฀ 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฀ curriculum฀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฀modeling฀ 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฀quantitative฀ and฀ functional฀ areas฀ to฀ allow฀
this฀to฀happen.฀The฀functional฀area฀faculty,฀including฀the฀administration,฀must฀
be฀ convinced฀ that฀ quantitative฀ literacy฀
is฀ invaluable฀ in฀ achieving฀ better฀ business฀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฀ piggyback฀on฀the฀overall฀plan฀for฀integration.
A฀ widely฀ acclaimed฀ integrative฀
approach฀that฀includes฀many฀of฀the฀elements฀ that฀ we฀ believe฀ are฀ important฀ for฀
July/August฀2008฀

373

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business฀students฀as฀part฀of฀the฀attempt฀to฀
improve฀quantitative฀literacy฀was฀developed฀over฀the฀last฀few฀years฀at฀the฀University฀of฀Arizona฀under฀the฀auspices฀of฀
a฀ multiyear฀ program฀ sponsored฀ by฀ the฀
Mathematical฀Association฀of฀America.฀It฀
is฀a฀two-course฀sequence฀that฀was฀developed฀and฀taught฀jointly฀by฀the฀university’s฀
business฀and฀mathematics฀faculties.฀The฀
courses฀cover฀probability฀and฀simulation฀
in฀part฀one฀and฀calculus฀and฀optimization฀
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฀ students,฀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฀population฀ 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฀universities฀and฀their฀business฀schools.
3.฀Quantitative฀ methods฀ courses฀ have฀
not฀ changed฀ much฀ in฀ half฀ a฀ century,฀
although฀ the฀ business฀ environment฀
has฀evolved฀dramatically฀with฀developments฀in฀computing,฀modeling,฀and฀
data฀collection.
4.฀Numeracy฀is฀just฀as฀important฀as฀lit-

374฀

Journal฀of฀Education฀for฀Business

eracy฀and฀should฀be฀similarly฀stressed฀
throughout฀the฀curriculum.
5.฀High฀ school฀ mathematics฀ followed฀
by฀ college฀ courses฀ in฀ calculus฀ and฀
statistics฀are฀insufficient฀for฀quantitative฀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฀ business฀ 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฀educators฀ to฀ remember฀ that฀ “for฀ most฀ students,฀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฀sciences฀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฀ University.
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฀
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