ÿWPC; Ì7í¶•]Ô±¤—¥O4„l›yQ¿: ¬³#ç˜cÄcjw‚íhܲ·/Šn¶Ýð8XOaû výDSÊ ÓÖÒv` ÀT7Ò¢4ÕÞzT#EKö¥p «”Ù2³ P¡£é>º*>\X‰JÞI¤A2†›ˆ-:ŽJ©¹öÜ%’lS6–^«åÂgÖ¿˜ùðg#f»3C‡'WóG½à^­fv(Ì*‘u‡¾×µR ˜å÷YS.¾`ØÐžÜ*´†C>ybaáJ æC‘QišiÀJf–³Ç•Œ r­ÏΚœêÑ|óY™sUµ1}©Ivò©oùÿ fæýºäwà°ûòLS\!\7TÔi]€‡¢yº¯%·§¥°-L¬9Œ²ïUú | Ò‡—bÏw@åv ¿vBrd»À/?6Fz‹øz†Õ‡MÚoÁ騶p^3N^dHñª€˜çz®ÔEQRJQ Ú­–.û,(×´±ÀÛÎe.H¦O™òF;±Zny#KýQuYM¨ž¿ åﵯv²¦Ÿ’Í@ïq½­.ó†·|¢^ž23’ïOÙFëê®õÀy%_—Σ¹- t‹€ #†ŒUN %` 0(fBŽw@Ð4$3˜Acrobat DistillerPSCRIPT,P<@@<,P0ÈÄ9W5®gHÅŒ @DÆw=(ÖÃ9 Z ‹6Times New Roman RegularX($¡¡C:\WINDOWS\TEMPLATE\STANDARD.WPT E  E #eéQ3|xÝ ƒ!ÝÝ  Ý›à@   ìàPROGRAMMATIC€REFORM:€THE€CARE€AND€FEEDING€OF€A€MATHEMATICS€MAJORˆÌÌà@MM"ìàG.€Daniel€CallonˆÌà@UU"ìàFranklin€CollegeˆÌÌÌMathematics€departments€across€the€country€(and€world„wide)€are€currently€at€various€stages€ofÏcurriculum€reform.€€Some€are€just€now€thinking€of€starting€implementation€efforts€(usually€at€theÏcalculus€level)€and€are€looking€at€various€existing€models,€while€others€have€a€decade€or€more€ofÏexperience€at€moving€beyond€the€traditional€lecture€format€and€content„driven€curriculum.€€InÏthe€latter€departments€reform€has€generally€spread€far€beyond€calculus,€reaching€from€upper„¼level€courses€to€service€and€developmental€courses.€€A€major€question€now€looms€before€suchÏdepartments:€what€comes€next?€€The€answer:€programmatic€reform.ÌÌTo€see€why,€letððs€examine€the€current€state€a€little€more€carefully.€€€Most€reform€efforts€haveÏbeen€centered€around€individual€courses,€using€one€or€more€of€the€following€approaches:€activeÏlearning,€applications„based€problem€solving,€cooperative€learning,€discovery€learning.€€TheÏincorporation€of€technology€generally€plays€a€central€role,€which€necessitates€students€(andÏfaculty)€learning€new€software€for€almost€each€course.€€Courses€are€designed€to€be€more€student„¼ññ¼¼ññcentered,€allowing€the€student€to€control€her€learning€process,€although€that€seems€to€be€more€aÏdream€than€a€reality.€€Many€aspects€of€these€reforms€seem€unique€to€a€given€environment€andÏinstructor,€with€variables€including€the€availability€of€technology,€the€types€of€students€in€aÏparticular€course,€and€the€departmental€atmosphere.€€It€is€rare€that€a€department€can€just€adopt€aÏset€of€available€curriculum€materials€and€head€to€the€classroom,€with€no€or€little€thought€as€toÏhow€to€adapt€them€to€that€particular€situation.ÌÌHence€the€need€for€programmatic€reform,€which€asks€two€major€questions:ÌÌà0  à*€How€do€we€blend€these€individual€changes€into€a€coherent€package?Ð (#(# ÐÌà0  à*€How€do€we€meet€the€needs€and€desires€of€the€students€and€stay€within€ever„tightening€à0@ "(#(#à¼ññ¼¼ññbudgets?Ð (#(# ÐÌIn€answering€these€questions,€especially€the€latter,€we€are€reminded€of€the€standard€warning€fromÏsystems€analysis€that€what€a€client€òòsays€he€wantsóó€is€usually€different€from€what€he€òòreally€wantsóó,Ð ¢#ò" Ðwhich€is€in€turn€different€from€what€he€òòneedsóó,€which€is€not€the€same€as€what€his€òòresources€willÐ ‹$Û# Ðallowóó.Ð t%Ä $ ÐÌMaking€the€process€of€programmatic€reform€more€difficult€is€the€changing€environment€of€higherÏeducation€in€general€and€the€teaching€of€mathematics€in€particular.€€Letððs€analyze€briefly€theÏcurrent€situation.€€ÌÌññÌññÌÌÐ ¥-õ(- Ð1.€€Higher€education€faces€a€changing€ò òclimateó ó.Ð ° ÐÌ*Parents€and€students,€and€consequently€administrators,€emphasize€value€for€the€dollar.€€ManyÏprivate€colleges€are€already€finding€students€making€college€choices€on€the€basis€of€whichÏinstitution€is€cheaper,€with€a€$100€difference€in€tuition€swaying€choices.€€We€seem€to€be€living€inÏa€Wal„Mart€society,€with€little€thought€to€the€consequences€of€our€decisions:€What€are€we€givingÏup€for€lower€costs?€€Will€lower€prices€today€mean€less€competition€(and€higher€costs)€tomorrow?€ÏAre€some€aspects€of€an€education€necessary€no€matter€how€expensive?ÌÌ*The€technological€explosion€makes€it€almost€impossible€to€keep€on€top€of€the€latestÏdevelopments.€€Few€institutions€can€afford€to€continually€purchase€state„of„the€art€equipment,Ïand€few€instructors€can€manage€to€keep€tabs€on€all€the€latest€software.€€Often€entering€freshmenÏhave€used€hardware€or€software€not€yet€available€at€their€college€or€university.ÌÌ*Our€society€is€saturated€with€information.€€Many€educators€are€convinced€that€colleges€andÏuniversities,€which€have€for€decades€(if€not€centuries)€seen€their€fundamental€role€as€teachingÏstudents€how€to€obtain€information,€will€have€to€adapt€to€a€new€paradigm:€teaching€students€howÏto€evaluate€and€judge€the€quality€of€information,€and€then€how€€to€draw€quality€inferences€fromÏthat€information.ÌÌÌ2.€€Higher€education€faces€a€challenging€ò òclienteleó ó.Ð Í ÐÌ*Students€and€parents€see€education€as€a€conduit€to€a€job€rather€than€as€an€important€goal€in€itsÏown€right.€€Choices€of€major€are€often€made€on€the€basis€of€how€many€employment€want€adsÏspecify€a€job€title€connected€with€that€major.€€Even€specific€courses€are€viewed€through€the€filterÏof€their€importance€on€a€resume.ÌÌ*Students€have€been€exposed€to€a€great€deal€of€technology,€but€their€knowledge€is€oftenÏsuperficial.€€They€may€be€aware€of€many€aspects€of€the€latest€innovations,€but€often€theirÏunderstanding€goes€at€most€to€the€level€of€a€casual€user,€so€that€even€minor€changes€may€result€inÏa€need€for€retraining.ÌÌ*Students€are€still€conditioned€under€the€old€paradigm€to€be€information„seekers.€€ObtainingÏinformation€is€considered€an€end€in€itself.€€Changing€ingrained€habits€takes€both€patience€and€aÏconscious€effort€to€motivate€students€to€make€such€changes.ÌÌÌ3.€€Mathematics€educators€face€a€changing€ò òcontentó ó.Ð F'–"& ÐÌ*Current€curricular€reforms€emphasize€innovation.€€As€long€as€the€pot€is€being€constantly€stirred,Ïitððs€hard€to€see€whatððs€cooking€and€how€well€done€it€is.ÌÌ*Current€curricular€reforms€rely€on€technology.€€Given€the€rapid€changes€in€technology,€this€isÏlike€learning€to€drive€using€a€different€car€each€day.€€While€the€underlying€principles€may€remainÏthe€same,€the€differences€may€distract€from€the€fundamental€goal.Ð ¥-õ(- Ї*Current€curricular€reforms€are€still€information„centered,€focused€on€the€old€paradigm.€€ManyÏquestions€remain€to€be€answered€regarding€how€much€and€what€type€of€information€is€needed€toÏform€an€adequate€foundation€while€still€working€on€the€development€of€understanding€ofÏmathematical€processes€(such€as€experiment„conjecture„proof)€and€lifetime€learning€andÏprofessional€skills€(including€communication€and€collaboration).ÌññÌññÌSo€with€such€shifting€sands€below€our€feet,€how€can€we€begin€to€implement€programmaticÏreform?€€At€Franklin€College,€we€have€already€taken€what€we€feel€to€be€the€first€steps€in€thatÏdirection.€€As€background,€let€me€mention€that€Franklin€is€a€small€private€baccalaureateÏinstitution€with€900€students,€45„50%€of€whom€are€first„generation€college€students,€many€ofÏthem€from€small€towns€in€Indiana.€€Incoming€freshmen€average€about€1000€on€the€old€SATÏ(before€recentering).€€The€Department€of€Mathematical€Sciences€has€five€full„time€and€oneÏcontinuing€3/4„time€faculty€positions€and€offers€majors€in€pure€mathematics,€mathematicsÏeducation,€applied€mathematics,€computer€science,€and€information€systems.€€We€have€been€atÏreform€over€seven€years,€with€the€first€formal€step€being€the€introduction€of€weekly€scheduledÏlabs€into€the€first€year€calculus€courses€in€1989.€€We€have€access€to€computerized€classrooms€onÏa€basis€ranging€from€once€a€week€for€our€liberal€arts€mathematics€class€to€every€day€for€suchÏheavy€users€as€numerical€analysis€and€multivariable€calculus.€€(This€access€illustrates€one€of€theÏadvantages€of€curricular€reform€in€small€institutions,€along€with€class€size€and€the€smaller€numberÏof€instructors€involved€in€teaching€any€given€course;€the€fact€that€our€teaching€loads€average€24Ïhours€per€year€with€up€to€six€preps€and€no€graders€available€provides€a€strong€downside€toÏattempts€to€keep€each€course€at€the€cutting€edge.)ÌÌThe€most€dramatic€and€rewarding€stride€we€have€made€in€programmatic€reform€is€to€develop€aÏset€of€four„year€goals€and€objectives€for€students€completing€a€major€in€mathematics.€€From€thisÏframework€goals€and€objectives€were€formed€for€each€course€and€sequence€of€courses€availableÏto€majors,€and€then€for€service€and€developmental€courses.€€Globally€these€goals€provide€a€basisÏfor€decisions€on€curriculum€requirements,€departmental€assessment€of€student€learning,€andÏdepartmental€initiatives€and€priorities;€locally€they€allow€instructors€to€make€informed€decisionsÏregarding€instructional€methods€and€content€within€individual€courses.€€The€goals€focus€on€threeÏgeneral€areas:ÌÌà0  à*Mathematical€concepts,€including€definitions,€theorems,€computations,€and€applications;Ð (#(# ÐÌà0  à*Mathematical€processes,€such€as€the€experimentation„conjecture„proof€cycle€or€theÏinteractive€dynamic€in€modeling€between€real„world€and€theoretical€results;Ð (#(# ÐÌà0  à*Lifetime€learning€and€professional€skills,€e.€g.,€communication,€collaboration,€and€use€of€à0@#(#(#à¼ññ¼ññtechnology.Ð (#(# ÐÌFrom€these€goals€we€produced€developmental€strands,€so€that€faculty€teaching€the€writing€ofÏproofs€€in€a€real€analysis€course€would€know€that€students€had€been€exposed€to€proof€concepts€inÏcalculus€and€that€the€experimentation„conjecture„proof€cycle€had€been€a€major€emphasis€in€theÏrequired€linear€algebra€class.€€In€this€way€we€can€assure€ourselves€that€primary€emphases€areÏaddressed€and€reinforced€in€a€conscious€manner.€€It€also€makes€it€easy€for€instructors€to€contactÏthose€who€taught€courses€earlier€in€the€developmental€strand€to€know€at€what€level€studentsÐ ¥-õ(- Ðshould€enter€their€courses.ÌÌOur€most€recent€step€has€been€programmatic€review,€examining€each€major€course€of€study€inÏmathematics€to€see€if€each€goal€and€objective€is€adequately€addressed€in€the€courses€required€forÏthe€major.€€We€also€looked€at€each€group€of€non„majors€taking€courses€in€the€department€to€seeÏthat€their€mathematical€needs€were€addressed€in€a€coherent€fashion.ÌÌOur€efforts€thus€far€have€been€guided€by€a€set€of€fundamental€operating€principles.€€They€can€beÏcharacterized€as:ÌÌÌ*ò òActive€learningó óÐ ³  ÐÌà0  àIf€education€is€to€have€any€real€lasting€benefit€in€preparing€for€a€lifetime€of€learning,Ïstudents€have€to€be€involved€as€more€than€passive€targets€in€the€learning€process.€€ActiveÏlearning€incorporates€a€broad€range€of€techniques,€including€discovery€learning€andÏcooperative€learning,€as€well€as€addressing€a€wide€variety€of€learning€styles.Ð (#(# ÐÌ*ò òAccessibilityó óÐ b ÐÌà0  àThis€incorporates€two€fundamental€ideas:€achieving€student€buy„in€to€course€objectivesÏby€effective€communication€of€the€rationale€behind€their€development,€and€aimingÏcourses€at€a€reasonable€level€for€where€students€are,€not€where€we€wish€they€were.Ð (#(# ÐÌ*ò òApplicabilityó óÐ ˆØ ÐÌà0  àThis€principle€likewise€has€two€facets:€demonstrating€connections€to€other€disciplines,€andÏhighlighting€relationships€to€potential€jobs€and€fields€of€graduate€study.Ð (#(# ÐÌ*ò òAssessmentó óÐ e ÐÌà0  àWe€have€to€be€innovative€to€devise€assessment€instruments€which€relate€to€our€goals€andÏobjectives,€many€of€which€cannot€be€adequately€evaluated€by€traditional€tests,€quizzes,Ïand€written€assignments.€€For€example,€in€our€senior€seminar€course€at€Franklin,€whichÏserves€as€a€capstone€experience,€we€have€supplemented€the€long„standing€hour„long€oralÏcomprehensive€exam€and€nationally„normed€multiple„choice€exam€with€a€ð ðjoint€writtenÏcomprehensive€examðð,€in€which€teams€of€four€to€five€students€spend€a€week€developingÏanswers€to€about€five€questions€which€require€modeling,€writing€proofs,€and€use€of€libraryÏand€information€resources.€€This€exam€is€written€and€graded€by€a€professor€at€anotherÏcollege€as€an€external€review€mechanism.€€Similarly,€new€teaching€methodologies€requireÏnew€assessment€tools.€€We€need€to€be€experimenting€with€a€wide€variety€of€assessmentÏinstruments€so€that€we€obtain€needed€feedback€and€also€so€that€we€can€steer€more€globalÏ(e.€g.,€university„wide)€assessment€efforts€in€directions€which€will€most€beneficial.Ð (#(# ÐññÌññÌÐ ¥-õ(- Ð*ò òAccountabilityó óÐ ° ÐÌà0  àWhile€many€see€accountability€as€a€dangerous€concept€to€be€feared€and€avoided€so€thatÏcolleges€and€universities€can€try€to€continue€to€be€the€judges€of€how€well€they€areÏsucceeding,€the€truth€is€that€the€climatic€changes€in€education€as€noted€above€willÏproduce€more€and€more€demands€for€accountability€from€boards€of€trustees,€fromÏstudents€and€parents€and€from€the€public€in€general.€€We€should€be€out€in€front€asking€toÏbe€held€accountable€so€that€we€have€a€voice€in€setting€goals€and€in€deciding€howÏaccountability€should€be€determined.Ð (#(# ÐÌÌIn€summary,€programmatic€review€means€asking€what€weððre€trying€to€do,€how€do€we€do€it,€andÏhow€do€we€know€if€weððve€done€it.€€It€means€looking€at€the€entire€undergraduate€and/or€graduateÏexperience€as€an€integrated€and€sequential€whole,€not€just€isolated€parts€at€specific€points€in€time.€ÏAnd€it€means€connecting€that€experience€to€the€world€outside€of€higher€education,€cognizant€ofÏwhat€happens€to€students€before€they€matriculate€and€what€happens€to€them€after€they€graduate.€ÏOnce€we€have€that€framework€in€place,€then€curricular€reform€can€achieve€its€full€potential€inÏtransforming€mathematics€education.