The Effects of a Graphing-Approach Intermediate Algebra Curriculum On

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The Effects of a Graphing-Approach Intermediate Algebra Curriculum on Students'Understanding of FunctionAuthor(s): Jeannie C. Hollar and Karen NorwoodSource: Journal for Research in Mathematics Education, Vol. 30, No. 2 (Mar., 1999), pp. 220-226Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/749612 .Accessed: 09/04/2011 14:44

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Brief Report

The Effects of a Graphing-ApproachIntermediate Algebra Curriculum onStudents' Understanding of FunctionJeannieC. Hollar,Lenoir-RhyneCollege

KarenNorwood,NorthCarolinaState UniversityIn this study,we extendedO'Callaghan'scomputer-intensive lgebrastudy by usinghis com-ponentcompetenciesandtheprocess-objectramework o investigate he effects of agraphing-approach urriculum mployingthe TI-82 graphingcalculator.We found thatstudents n thegraphing-approachlasses demonstratedignificantlybetterunderstandingf functionson all4 subcomponentsof O'Callaghan'sFunctionTest, includingthe reificationcomponent, handid studentsnthetraditional-approachlasses.Additionally, osignificant ifferenceswerefoundbetween hegraphing-approachndtraditional lasseseitheron a finalexamination f traditionalalgebraskills or on an assessmentof mathematicsattitude.Key Words:Algebra;Conceptualknowledge;Curriculum; unctions;Graphing alculators

The functionconceptis consideredby manyto be one of the most centralcon-cepts in all mathematics,yet it is one for which studentsrarelydevelopadequateunderstanding.Numerous studies (Harvey, Waits, & Demana, 1995; Mayes,1995; Ruthven, 1990) have been conductedto examine the effects of graphingtechnologies and graphingcurriculaon function understanding.These studieswere generallyfocused on student achievementand generallyshowed that stu-dents using graphing echnology performedas well on tests of traditionalalge-bra as didstudentswithout suchtechnologyand,atthe sametime,improvedper-formanceon visual andgraphing asks.Kieran(1993) called for a different ypeof research n which we

domore han imply how hat echnologyanhelpourstudentsolearn uchmate-rial.We have o think bout ommittingurselvesoverydetailed tudies f humancognitionn this domain.Theprocess/objectheoreticalrameworkan,attheveryleast,helpto tiesuch indings ogether.p.223)By process/object rameworkKieran was referring o the ontologicaldualityoffunctions;thatis, a functioncan be thoughtof in two ways: operationally as aprocess)andstructurallyas an object).

Thisarticles basedonthefirstauthor'sissertationompletedtNorthCarolina tateUniversityunder he directionof the second author.


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One researcherwho did incorporate process-objectduality rameworknto hiswork was O'Callaghan 1998). O'Callaghan tudiedthe effects of the Computer-IntensiveAlgebra(CIA)(Fey, 1992)curriculumncollege algebra tudents'under-standingof the functionconceptby comparing tudentsusingthe CIA with studentswho experienceda traditional urriculum.He developeda function est,diagnosticin nature, o assess students'understandingf functions.Eachquestionon the testwas designedto assess one of the following aspectsof conceptualknowledgeoffunctionswithout he use of the graphing alculator:a) modelinga real-world it-uationusinga function, b) interpreting function n termsof a realisticsituation,(c) translating mongdifferentrepresentationsf functions,and(d) reifyingfunc-tions.Reification efers o thetransitionrom theoperationalo thestructuralhaseof conceptdevelopment.O'Callaghan1998) foundthat he CIA studentsachieveda betteroverallunderstandingf functions hantraditional tudentsand were betteratmodeling, nterpreting,ndtranslating,uthe foundnodifferencesorreification,the most difficult of the four aspects of function knowledge. In addition,O'Callaghanadministeredhe Revised MathematicsAttitudeScale (Aiken, 1972;Dutton,1962)andfoundthatstudentsnthe CIA curriculumignificantlymprovedtheirattitudes owardmathematics ver the semester.Our main purposein this study was to extend O'Callaghan'sCIA study byusinghis frameworko investigatea differentcurriculum,pecificallya graphing-approach urriculum mployingthe TI-82 graphingcalculator.We wantedto seeif his results on the four componentsof his function test and on the RevisedMathematicsAttitudeScale wouldhold withina differentcurriculumncorporat-ing graphingcalculators.Of particular nterest was the reificationcomponent.Kieran(1992) reported hatalthoughstudentsrarelyacquireanyreal sense of thestructural spectsof algebra,graphingsoftwaremight help to develop structuralconceptions.We wanted o determinewhether hegraphing urriculum longwiththe graphingcalculator acilitatedreificationof the functionconcept.In addition,we wantedto examine studentperformanceon a test of traditional lgebraskillsto determine he influenceof the graphingcurriculum.

METHODOLOGYParticipants

The participants n the study were college studentsenrolledin intermediatealgebra.The samplewas taken from studentsenrolledin the intermediatealge-bra course at a large state university with approximately28,000 students.Studentsenrolledin intermediatealgebrawere those studentsscoringlowest onthe university'smathematicsplacementexamination.A total of 90 studentspar-ticipated n the study:46 in the treatmentgroupand44 in the controlgroup.Treatments

The college text Intermediate Algebra: A Graphing Approach (Hubbard &Robinson,1995) was used in conjunctionwith TI-82sin the treatment lasses. A


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balance of graphingcalculatorand traditionalalgebrawork is foundin the text,which includes explorationand discovery examples to help guide studentstolook for patternsand make discoveries. Use of the TI-82 enabled students toexplore, estimate, and discover graphicallyand to approachproblemsfrom amultirepresentationalerspective.The studentshad access to the calculatorsbothin class and for homework exercises and tests but not for the O'CallaghanFunctionTest or the traditional inal examination.

The textbook used in the control classes, Intermediate Algebra: Concepts andApplications, ourthedition(Bittinger,Keedy,& Ellenbogen,1994), coveredthesame topics as the experimental ext but emphasized memorizing solated factsand proceduresand becoming proficientwith paperand pencil calculations.Itfocused on simplifyingandtransforming xpressionsandsolvingequations.Thecontrolgrouphad no knowngraphingcalculatoraccess.Instrumentation

The function test developed and used by O'Callaghan (1995, 1998) wasadministered o all students n this studyboth as the pretestat the beginningofthe semester and as a posttestat the end of the semester.The test is designedtobe administeredwithoutaccess to graphingcalculators.It is diagnosticin naturein thateach questionis designedto assess one of the following aspectsof con-ceptual knowledge of functions:(a) modeling a real-worldsituation,(b) inter-pretinga function n termsof a realisticsituation, c) translating mongdifferentrepresentations f functions,or (d) reifyingfunctions.The instrument hosen to evaluate students'traditionalalgebraskills was thedepartmentalinalexamination,a 50-questiontestof conventionalalgebraskills.It was administeredo all four classes duringthe final week of the semester,andstudentswere allowed 3 hours to completeit.To measure students' attitudes toward mathematics,we administeredtheRevised MathematicsAttitude Scale (Aiken, 1972; Dutton, 1962) both beforeandafter reatment, s didO'Callaghan.This instrumenthasbeen usedin numer-ous studies and is considered o be one of the measuresof choice regardingatti-tudes towardmathematics.Research Design

The performanceof studentswho hadused the TI-82 in a graphing-approachcurriculumwas comparedwith the performanceof studentswho had been in atraditionalalgebracurriculum,using the instrumentsdescribedabove.Four sec-tions of a semester-longintermediatealgebracourse were used in a balanceddesign with two instructorseach teaching one experimentaland one controlclass. One of each of two simultaneousmorningsections and two simultaneousafternoonsections were randomlyselected to use the experimentalcurriculum.Studentsregistered or classes by telephonevia a computerized cheduling pro-gram.Classpopulationswere expectedto be similar. Students n the experimen-


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Jeannie C. Hollar andKarenNorwood 223

tal sections were given the opportunity o switch to the control section beingtaughtat the sametime, but none elected to do so.Theresearchers bservedeachof theexperimental ndcontrolclasseson a ran-dom basis throughout he semester,focusing mainly on teachers' behaviorandlesson developmentandon students'behaviorand calculatoruse. For each treat-ment, the instructorsplanned together and followed the same plans of study,adhering o the coursesyllabus.From nterviewsand observations heresearchersconcludedthat the instructorswere not biasedin favorof eitherapproach.Descriptivestatisticsandanalysisof variance ANOVA)procedureswere per-formedon both theO'Callaghanunctionpretestresultsand thedemographic ari-ables to determineanyinitialdifferencesamongthe four classesin thestudy.Classprofilesof typicalcharacteristics rovideddata about students'genders(as mea-suredby thepercentagehatweremales),averageages,mathematical ackgrounds(numberof previousalgebracourses),ability n mathematicsMathSAT scores),predictedgrade-pointaverage n mathematics PGM)calculatedby departmentalformula,and verbalability(VerbalSAT scores).Analysisof these characteristicsindicated hatthe students n the four classes were differentonly with respecttogendercomposition.Thereweremorefemale studentsoverallandmoremales thanfemales in theexperimental roup.Analysisof the functionpretestscores indicat-ed no significantdifferencesamongthe fourclasses on priorknowledgeof func-tions,whichthereforewas notused as a covariant n the finalanalysis.

RESULTSO'Callaghan Function Test

Students'understanding f the functionconceptwas analyzed nitiallyusing aMANOVA on the fourcomponentscores and the total scoreon the functiontest,and this analysiswas supplementedby univariateresults on the individualcom-ponents.Table 1 shows the means andstandarddeviationsby treatmentor eachcomponentandfor the total score.

Table 1Function Posttest Mean ScoresExperimental Control

Component Maximumscore M SD M SDModeling 7 4.32 1.65 3.33 1.64Interpreting 11 7.46 1.92 5.90 2.21Translating 9 5.05 2.26 3.64 2.21Reifying 10 4.20 1.89 2.74 0.29Total 37 21.02 5.87 15.62 4.70

The experimentalclasses hadhighermeans for each of the fourcomponentsas well as for the total score on the functiontest. TheMANOVA,F(4, 69) = 4.68,revealed an overall significanttreatmenteffect at the a = .01 level, indicating


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224 Effects of a Graphing-Approach urriculum

thatthe experimentalclasses had a significantlybetteroverallunderstanding ffunctionsthanthe controlclasses had. Univariateresultsof ANOVAs on thefourcomponentsrevealedsignificantdifferences at the a = .05 level in favor of thegraphing-approach roupfor modeling, interpreting,andtranslating,and at thea = .01 level for reifying.Departmental Final Examination

Althoughthe experimentalgroupoverall had a slightly highermean score onthis examination usedto measure he students' raditional lgebraskills),no sig-nificant difference was found between the scores of treatmentand controlclass-es. Similarly,althoughone instructor's tudentshad a slightly highermean scorethantheother nstructor's tudents,no significantdifferenceswere found for maineffects of instructor r genderor for anyinteractionsamongthe three variables.Attitude Survey

The attitudesurveyposttest,conductedafter studentshadcompletedthe alge-bra course except for the departmentalinal exam, showed thatstudents n theexperimentalclass hadslightly morepositive attitudes han theircounterpartsnthe controlclass had about mathematicsand theirmathematicalabilities.Therewas, however, no significantdifference between the experimentaland controlclasses, betweeninstructors,or betweenthe genders.

CONCLUSIONSAND DISCUSSIONFunction Test

Becauseof theavailabilityof graphingcalculators, he graphing-approachur-riculum can include examplesandproblemsfor modelingreal-worldsituationswith functionsthat wouldbe either too time-consumingor impracticalwithoutagraphingcalculator.The graphingcalculatoraffords the userboth the abilitytocreateequations,tables, andgraphs quicklyandthe facility to move amongtherepresentations apidly.Thus, it can be concluded that the graphing-approachstudentswho used the TI-82 weremorecomfortable hanthe traditional tudentswhen workingwith real-worlddata and situations.The experimentalgrouphadbecomeaccustomedoverthe semester o examiningfunctionsfrom differentper-spectives andaccordinglyperformed ignificantlybetterthanthe traditional tu-dentson interpreting ndtranslatingquestions.The resultson thefirst threecom-ponentsare in agreementwith O'Callaghan's indingswith the CIA curriculum.On the reification component, however, we found a significant differencebetween the graphing-approach group and the control group, whereasO'Callaghan oundno difference.The reificationcomponentscores,whichwerethe lowest of anyfor the fourcomponents orboththe traditional ndexperimen-tal groups in both studies, indicate the difficulty of the reificationprocess, a


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process that involves a much greaterdegree of abstraction hanthe otherthreeaspects of functionknowledge. Reification is not a process that can be taught.Instead, t is the shift involved in makingthe transition rom an operational o astructural nderstanding f a concept.In O'Callaghan's tudyof the CIAcurricu-lum, studentshad access to graphing echnology only in a lab setting.The graph-ing-approach tudents n the present studyhad access to the graphing-calculatorduringevery class meetingas well as for homeworkand so had moreopportuni-ties to explorefunctions andto examineabstractapplications.This difference naccessto graphing echnology mayaccount orthe differences n reification oundfor the two groups n this studybutnot foundin the O'Callaghan tudy.No significantdifferences in traditionalalgebraicskills were found betweenthe experimentalandcontrolgroups.It can thereforebe inferred hat the experi-mental studentswho used graphingcalculatorsfor the semesterwere not hin-deredin theircomputationalability.The graphing-calculatorreatmentwas notexpectedto give the studentsany advantageover the traditional tudentsbecausethe final examination focused mainly on paper-and-pencilcalculations andmanipulationssuch as simplifying and transforming ymbolic expressionsandsolving equations.These results are somewhat congruentwith the findings inO'Callaghan's(1998) study. Although O'Callaghan ound a lower level of pro-ficiency in symbolicmanipulationwith the CIA students hanwith his tradition-al students,he found no significantdifferences afteradjustingfor the CIA stu-dents' initial lower mathematical ompetence.We found that students n the graphing-approachurriculum s a groupdid notdiffer from traditional students in their attitudes toward mathematics, butO'Callaghan(1998) found that studentsusing the CIA curriculum ignificantlyimprovedtheir mathematicalattitudes over the course of the semester.As yet,there is no consensus on the effect of technologyuse on attitude.Recommendations for Further Research

Ithas been argued hat"ofall the areaswhere further esearchcouldprofitablybe carriedout, the one that seems to stand out as being clearly in need of atten-tion is that of finding ways to develop structuralconceptions in students"(Kieran,1992, p. 413). A continued researchfocus is needed to help find waysto facilitate the transition rom operational o structural onceptionsin students.Researchon the reificationof functionsand otherconceptsshould be expanded.Studies are needed to advancethe knowledge of how structural ndproceduralconceptions interact when students are doing algebra within a technologicalenvironment.It is important o studyhow technology positively and negativelyaffects the developmentof both structural ndprocedural onceptions.REFERENCES

Aiken,L. R., Jr.(1972). Researchon attitudesn mathematics.TheArithmeticTeacher,19, 229-234.Bittinger,M. L., Keedy, M. L., & Ellenbogen,D. (1994). Intermediate lgebra: Conceptsandappli-cations (4th ed.). New York:Addison-Wesley.


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Dutton,W. H. (1962). Attitudechangeof prospectiveelementary chool teachers owardarithmetic.TheArithmeticTeacher,9, 418-424.Fey, J. T. (1992). Computer-intensive lgebra: An overview of fundamental mathematical andinstructional hemes.Unpublishedmanuscript.Harvey,J. G., Waits, B. K., & Demana,F. D. (1995). The influence of technologyon the teachingandlearningof algebra.Journalof MathematicalBehavior, 14, 75-109.Hubbard,E., & Robinson,R. D. (1995). Intermediatealgebra: A graphing approach. Lexington,MA: D. C. Heath.Kieran,C. (1992). The learningandteachingof school algebra.In D. A. Grouws(Ed.),Handbookofresearch on mathematics eachingand learning(pp. 390-419). New York:Macmillan.Kieran,C. (1993). Functions,graphing,andtechnology:Integrating esearchon learningand instruc-tion.InT. A. Romberg,E. Fennema,& T. P. CarpenterEds.),Integratingresearch on thegraph-ical representationof functions (pp. 189-237). Hillsdale,NJ:Erlbaum.Mayes, R. L. (1995). The applicationof a computeralgebrasystem as a tool in college algebra.School Science andMathematics,95, 61-68.O'Callaghan,B. R. (1995). The effects of computer-intensive lgebraon students'understanding fthe function concept (Doctoral dissertation,Louisiana State University, 1994). DissertationAbstractsInternational,55, 3440A.O'Callaghan,B. R. (1998). Computer-intensive lgebraand students'conceptualknowledgeof func-tions. Journal or Research in MathematicsEducation,29, 21-40.Ruthven,K. (1990). The influence of graphiccalculatoruse on translation romgraphic o symbolicforms. EducationalStudiesin Mathematics,21, 431-450.

AuthorsJeannie Hollar, AssistantProfessor,Departmentof Mathematics,Lenoir-RhyneCollege, P.O. Box7208, Hickory,NC 28601; [email protected] Norwood, Associate Professor, College of Education and Psychology, DepartmentofMathematics,Science,andTechnologyEducation,NorthCarolinaStateUniversity,326 Poe Hall,Box 7801, Raleigh,NC 27695-7801; [email protected]

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The Effects of a Graphing-Approach Intermediate Algebra Curriculum On