The Abstraction Process of Limit Knowledge · 2017. 10. 17. · 347 Sezgin Memnun, Aydın,...

Received: July 15, 2016Revision received: September 28, 2016Accepted: November 3, 2016OnlineFirst: December 30, 2016

Copyright © 2017 EDAMwww.estp.com.tr

DOI 10.12738/estp.2017.2.0404 April 2017 17(2) 345–371

Research Article

KURAM VE UYGULAMADA EĞİTİM BİLİMLERİ EDUCATIONAL SCIENCES: THEORY & PRACTICE

1 Correspondence to: Dilek Sezgin Memnun (PhD), Department of Elementary Education, Faculty of Education, Uludag University, Nilüfer, Bursa 16059 Turkey. Email: [email protected]

2Department ofMathematics Education,AhmetKeleşoğlu Faculty of Education, Necmettin ErbakanUniversity,Meram,Konya42090Turkey.Email:[email protected]

3DepartmentofTarsusVocationalHighSchool,MersinUniversity,Tarsus,Mersin33400Turkey.Email:[email protected] of Educational Sciences,AhmetKeleşoğlu Faculty of Education,Necmettin ErbakanUniversity,Meram,Konya

42090 Turkey. Email: [email protected]

Citation:SezginMemnun,D.,Aydın,B.,Özbilen,Ö.,&Erdoğan,G.(2017).Theabstractionprocessoflimitknowledge.Educational Sciences: Theory & Practice, 17,345–371.http://dx.doi.org/10.12738/estp.2017.2.0404

Abstract

The RBC+C abstraction model is an effective model in mathematics education because it gives the opportunity

to analyze research data through cognitive actions. For this reason, we aim to examine the abstraction process

of the limit knowledge of two volunteer participant students using the RBC+C abstraction model. With this

aim, the students’ abstraction processes were examined over recognition, building-with, construction, and

consolidation. In the process, three problems that enable students to use their pre-knowledge were designed

by the researchers to reveal the students’ mathematical thinking levels, abstraction processes, and finally to

improve a new structure. These problems were used in the application, and students’ dialogues were video-

recorded. The semi-constructed interview and observation data-collection methods were used in this process,

and one of the researchers participated in the study as a participant observer. The data was analyzed and

interpreted by transcribing the video recordings into written text and grouping them into the cognitive actions

of the RBC+C model. In conclusion, these students were indicated to have managed recognizing and building-

with their pre-knowledge on sequence, function, and infinity, and thus constructing knowledge on limits.

Keywords

Abstraction • Building-with • Construction • Limit • RBC+C Abstraction model

Dilek Sezgin Memnun1

Uludag UniversityBünyamin Aydın2

Necmettin Erbakan University

Ömer Özbilen3

Mersin University

Güneş Erdoğan4

Necmettin Erbakan University

The Abstraction Process of Limit Knowledge

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EDUCATIONAL SCIENCES: THEORY & PRACTICE

Limitsareabasicandimportantconceptinmathematics.Theycanbeassumedas one of the most fundamental concepts and influential instruments of generalmathematics because they lead especially to theunderstandingof derivations andintegrals (Çıldır,2012; Çoker,Özer,&Taş,1989).Inotherwords,limitknowledgeisa pre-concept related to many important mathematical concepts such as derivatives, integrals, continuities, and convergences. Therefore, it also constitutes an important role in high school and college mathematics curricula.

Knowledge and Historical Development of LimitsLimitswereusedintheoremsrelatedtoshapeswithcurvilinearsidesinearlytimes.

EuclidandArchimedesusedthisconceptforthefirsttime.Forexample,Archimedesusedthemtocalculatetheareaofacircle.Inhisjustification,theareaofthecircleisequaltotheareaofarighttrianglewhosebaseequalsthecircle’scircumferenceandwhoseheightequalsthecircle’sradius.Inthisprocess,thecircleisassumedtobeastraightlineequaltothecircumferenceofit.Hecalculatedtheinferiorandsuperiorlimits tofind the straight line in this justification (Cajori, 2014, p. 25).However,theuseoflimitswiththesamecurrentmeaningwasbasedonworkfromthe17thcentury.Forexample,Fermatexaminedtheinterrelationbetweenthelimitofacurveat a point on the graph and the tangent of that point of the curve (Baki,2008,p.147). Afterwards,theresearchersNewtonandLeibnizusedlimitstocalculateintegrals.Inotherwords,differentialequationsemergedfromknowledgeonlimits(Baki, 2014, p. 145).Newtoncanbesaidtohaveusedthisconceptingeneral,notforanyspecificpurpose.Heused it to extrapolate a fewdifferentprinciplesused for avarietyofpurposes (Cajori,2014,p.234). Similarly, the area under the curve of y = x2 restricted for 0 < x<1wasshowntobe1/3byCavalieriusinglimitknowledge(Baki,2008,p. 146).Meanwhile,thedefinitionofthelimitwasnotgivenuntil1817byBolzano.Additionally, Weierstrass and Cauchy indicated limits were applied in calculusnineteenth century (Arslan&Çelik,2015,p.483;Baki,2008,p.149). Formal and informaldefinitionscanbegivenforlimitknowledge.Incalculus,theepsilon-deltadefinitionoflimitisaformalizationofthenotionoflimit(Balcı,2014).

Difficulties in Learning and Teaching LimitsThesedays,difficultiesandmisconceptionsaboutlimitknowledgeareencountered

by students as learning the subject of limits brings along serious difficulties forstudentsintheadvancedsubjectsofhigh-schoolcurriculum.Becauselimitscontaindifferent procedures that include infinity in particular, they are also not a simplesubject(Baştürk&Dönmez,2011;Tangül,Barak,&Özdaş,2015). Most students cannotplacetheknowledgeoflimitsintheirmindsandforthisreasonhavedifficultymakingsenseofit;asaresult,studentshavetroublelearninglimitknowledge(Çıldır,

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Sezgin Memnun, Aydın, Özbilen, Erdoğan / The Abstraction Process of Limit Knowledge

2012).Specifically,theyhavedifficultyperformingoperationsrelatedtolimits.Thissituationhasbeenreflectedindifferentstudies.Withinthiscontext,Durmuş(2004); Gürbüz,Toprak,Yapıcı,andDoğan(2011); and Tatar,Okur,andTuna(2008) aimed to determinedifficultsubjectsinmathematicscoursesandtorevealtherootsunderlyingthesedifficulties.Thusallof thesestudiesfoundthesubjectof limits takingplaceatthetopofsubjectswhichstudentsperceiveasdifficult.Thedesiretouncoverthedifficultieswithlimitsisthemainreasonwhythenumberofstudiesmadeonthisconcept has increased (Baştürk&Dönmez,2011).

Somelimit-knowledgestudies(Barak,2007;Baştürk&Dönmez2011;Denbel,2014; Juter, 2006; Szydlik, 2000; Tangül et al., 2015)revealedstudents’difficultiesandmisconceptions.InthestudiesbyBarak(2007) and BaştürkandDönmez(2011), studentsweredeterminedtohavedifferentmisconceptionsaboutlimitknowledge.In this context, Barak (2007) explained these misconceptions as errors and misconceptionsabouttheepsilon-deltadefinition,thedefinitionoflimits,afunction’slimitatapoint,limitsapproachingfromtherightandtheleft,therelationbetweentheconceptsoflimitsandcontinuity,afunctionthatisdefinedatasinglepoint,graphingfunctions, understanding the concept of infinity, and using the limit in equations.Additionally, BaştürkandDönmezsummarizedpre-serviceteachers’misconceptionsthatlimitshouldbedefinedandextendfromafunctionatasinglepointwhilebeingthelimitofafunctionatthesamepoint.Similarly,limitsdonotexistwhenadiagramisnotcontinuousorwhenthereareomissions/fractionsinit.Denbel(2014) indicated thatuniversitystudentsseealimitasunreachable,anapproximation,aboundary,ora dynamic process. Also, Denbeldeclared that students have many misconceptions, suchaslimitsrepresentingthevalueofafunctionatapoint,alimitcanbefoundthough a method of substitution, or that one divided by zero equals zero. Juter (2006) reported that students havemany different cognitive difficulties related tolimitknowledge,suchasinfiniteseriesandhowfunctionsneverreachtheirlimits.Szydlik (2000) declared that limit beliefs and understanding are highly related toone another. Similarly, Tangül et al. (2015) stated that most university students have troubleexpressinglimitsandusingitsformaldefinition.

Methods for Teaching and Learning LimitsSome studies (Akbulut&Işık,2005;Akkoyunlu,Güler,Uğurel,&Alan,2003;Biber

&Argün,2015;Bukova-Güzel,2007;Çıldır,2012;Dönmez&Baştürk,2010;Kula&Bukova-Güzel,2015a,2015b;Przenioslo,2004;Roh,2007;Tangületal.,2015) have investigatedhowstudentslearnlimitsandhavegivenroomtosometeachingstudieswith theaimofeliminating theirdifficultiesandmisconceptions in relation to limitknowledge.AsaresultofAkbulutandIşık’s(2005)study,anewinteractiveteachingstrategywasdeclaredtobeinfluentialonteachinglimitknowledge.Akkoyunlu et al.

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(2003)explainedthatanimations,activities,scenarios,andworksheetscanbeusedascoursecontenttoclarifyhigh-schoolstudents’limitknowledgesoastoovercometheirdifficulties. In this study,misconceptionsarealso statedaspreventableby focusingon how a function’s appearance behaveswhen the function’s independent variableis infinitely small. Biber andArgün (2015) indicated that university students have structuredlimitknowledgeondoublevariablefunctionsmostlybymakingexpansivegeneralizations.Theyalsoexplainedthatcandidatesarenotsuccessfulenough,whichrequires reconstructing their generalization processes. Çıldır(2012) declared that using computer-baseddemonstrationscouldbebeneficialforteachinglimitsandovercomingrelated misconceptions. Dönmez and Baştürk (2010) declared that current teacher-trainingprogramsareinsufficientandimproperatteachingpre-serviceteachershowto integrate different strategies. Kula and Bukova-Güzel (2015a) declared that pre-service teachers’ thoughts about currentmathematics programs related to limits arecomposedundertheheadingsofrelatingreallifetolimitknowledge,relatingthemindifferentareas,providingmathematicalthinkingandreasoning,improvingtheabilitytocommunicate,usingmathematicallanguage,relatingmathematicswithart,andusingtechnology effectively. Kula and Bukova-Güzel (2015b) investigatedwhich specialmethods pre-service teachers use in mathematics for limits. They indicated that pre-serviceteachers’preferredmethodsforteachinglimitsarelimit-specificrepresentationsand activities. Their participants used mostly verbal and algebraic representations.Przenioslo (2004) declared that students generally interpret the formal definitionoflimit as an equation. Roh(2007)explainedthatafemalestudentwhowasengagedintheε-stripactivityunderstoodasawholetheconceptandformaldefinitionofthelimitofasequence. Tangül et al. (2015) explained that most university students do not interrelate conceptimagesandformaldefinitionsbutuseconceptimagestosolvelimitproblems.Moreover, Bukova-Güzel (2007) reported that mathematics education departments promotelearninglimitknowledgethroughaconstructivistlearningenvironment.

Abstraction, the Abstraction Process, and the RBC+C Abstraction ModelAlong with attaching more importance to conceptual learning in these renewed

programs, abstraction has also gained importance. Conceptual learning entails theabstractionofaconcept.Forthisreason,suchmattersasconstruction and abstraction, which requireexamining individuals’ abstractionprocesses,havebecome importantresearchsubjectsinthelearningdomainofmathematics.Moreover,mathematicsisascienceofabstraction,andthegreatmajorityofmathematicalconceptsareobtainedthroughabstraction(Altun, 2014, p. 5). This makes it especially important to overcome theabstractionprocessofconcepts inmatheducation.Becauseof this,determininghowtodevelopabstractionsandgeneralizationsthatstudentscarryoutwiththehelpoftheirforeknowledgebecomesimportant(Bukova-Güzel,2006,2007). As a result, this studyexamineshow12th-gradestudentslearnandabstractlimitknowledge.

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Abstraction is a structural process, the construction of mental structures frommathematicalstructures,aswellasmathematicalstructuresfrommentalstructures.Itistheprocessofdetachingacharacteristicorcharacteristicsthatobjectshaveincommonwithotherobjectsandgivingnamestothesecharacteristics.Theabstractionprocess isnot adirectlyobservable state. It takesplace in the formof isolatingaconcept from its specific characteristics and directing the process from a set ofcontexts toward a concept (Dreyfus, 2007;Mitchelmore, 2002; Sierpinska 1994;Tall, 1988;Yılmaz, 2011).Also, mathematical abstraction provides opportunitiesforconstructingmathematicalconcepts,aswellasachievingthetransitionbetweenconcepts and, most importantly, comprehension not through memorization butthrough conceptualization (Can, 2011). For this reason, the theme of abstractionoccupiesanimportantplacethroughoutone’smathematicaleducation.

Differentmodelshavebeendevelopedandusedforabstraction.OneoftheseistheRBC+C(recognizing,building-with,construction,andconsolidation)abstractionmodel. According to this model, one needs to rearrange old structures in order to reach a new structure, establish connections and relationships between them, andintegratethemintoasinglethoughtprocess.Thismodelallowsanalyzingstudents’knowledgeconstructionthroughthecognitiveactionsofrecognizing, building-with, construction, and consolidation.Inthisabstractionprocess,abstractionisformedbypassingthreesteps.Thefirstistheneedforanewstructure.Thesecondistheformationofanewabstractentitythatprocessestheactionsofrecognizingandbuilding-withasexistingstructuresareintermingled.Thethirdoneisconsolidatingabstractioninsuchawayastofacilitateaperson’srecognition.Additionally,abstractionoccursinaproblem-solvingprocessinwhichthestudentperformstheactionofformationusinga newmethod (Hershkowitz, Schwarz,&Dreyfus, 2001). While forming a piece ofknowledge,anindividualmakesassociationswithprevioussubjectmatterintherecognizingstage,usesitforproblem-solvinginthebuilding-with stage,setsupanewknowledgestructureintheconstructionstage,andreinforcestheformedknowledgeintheconsolidationstage.Abstractionemergesbyrequiringfurtherconstructionandalso covers strengthening (Tsamir&Dreyfus,2002). The act of construction forms as a result of the materialization of recognizing andbuilding-with,whicharenestedin one another (Dreyfus,2007;Dreyfus,Hershkowitz,&Schwarz,2001b). Thus, the RBC+Cmodelisimportantbecauseitenablesonetoexaminetheabstractionprocessover cognitive ones (Dreyfus, 2007; Dreyfus, Hershkowitz, & Schwarz, 2001a,2001b;Herskhowitzetal.,2001).Eachoftheseactionsisobservable;theabstractionprocesscanberecognizeddeeplybyobservingthem(Altun&Yılmaz,2008) and is appropriate for this research.

Thisabstractionmodelhasbeenexaminedusingvariousmathematicalconceptsin different studies: the greatest integer function (Altun&Yılmaz,2008), algebra

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and properties of arithmetic operations (Dreyfus et al., 2001a, 2001b), ratios (Hassan &Mitchelmore, 2006), and probabilities (Dreyfus, Hadas, Hershkowitz,&Schwarz,2006;Hershkowitz,2004;Schwarz,Dreyfus,Hadas,&Hershkowitz,2004). In addition, functions (Hershkowitzetal.2001),absolutevalue(Özmantar,2004, 2005), linear equations (Sezgin-Memnun&Altun,2012),infinity(Tsamir&Dreyfus, 2002),andtrianglesalongwithitsvariations(Yeşildere&Türnüklü,2008a,2008b)havebeenexploredinparticular.

Limit Knowledge in Turkish Mathematics Course ProgramsLimitknowledgehadbeentaughtintuitivelywithoutaformaldefinitiongivento

students prior to the changes in mathematics course programs in Turkey in recent years (Ministry of Education [MEB], 2006). After the changes in the programs, constructivistlearningfirstbeganitsuseinprogramsforallmathematicalsubjects,andalsothesubjectoflimitshasoccurredindifferentgradesinmathematicsprogramsinrecentyears.Finally,thesubjectoflimitshasninedifferentacquisitionsandtakestwelvehourstoteachandlearn(MEB,2013) per the latest version of the reorganized programs. Also, the acquisition of limits is explained in the course programs. Some of these are:

1.Explainanindependentvariableapproachingagivennumberwithexamples.

2. Explain limits at a point of a function, the limit approaching from the right or left,withexamples.Oneshouldexplaintherelationbetweenalimitapproachingfrom the right and one from the left.

3.Explainthepropertiesoflimits;makeexercisesforthem.

4. Make exercises related to the limit of functions.

Whichadditionalstrategiescanbeappropriateforteachinglimitknowledgehasnot been detailed (Kula & Bukova-Güzel, 2015b;MEB, 2013). Because of that, differentstrategieshavebeenattemptedindifferentstudies(Akbulut&Işık,2005;Akkoyunluetal.,2003;Bukova-Güzel,2007;Çıldır,2012).

ObjectivesDifferent studies have emphasized the importance and need to research the learning

processoflimits,whichhaveanimportantplaceinmathematics.Also,studentshavetroubleconstructinglimitknowledge.Becauseof this,somestudies(Barak,2007;Baştürk&Dönmez2011;Denbel,2014;Juter,2006;Szydlik,2000;Tangületal.,2015)haveinvestigatedstudents’misconceptionsaboutlimits,andothershavetriedtosolvethemwithnewteachingandlearningstrategies.Forexample,AkbulutandIşık

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(2005) tried to teach the concept using an interactive teaching strategy, Çıldır(2012) used computer-generated visualization, Bukova-Güzel(2007)applied constructivist learning, and Akkoyunlu et al. (2003) used animations, activities, and scenarios to eliminate students’ difficulties andmisconceptions. Similar to Bukova-Güzel’s(2007) research, yet different from all previous studies, the constructivist learning approach has been used in this research, as it is appropriate for abstracting limitknowledge.Abstracting limit knowledgehas been explored through constructivistlearningusingtheRBC+Cabstractionmodelinthisresearch.Limitknowledgewaschosenforthisresearchduetothesubject’scomplexityandabstractconcepts(Çoker etal.,1989).LimitknowledgeisalsoappropriateforcarryingoutthisstudyusingtheRBC+Cabstractionmodelbecauseonecomponentoftheabstractionprocessisto have a mathematical entity (Özmantar&Monaghan,2007).Additionally,veryfewstudiesrelatedtolimitknowledgehavefocusedonhigh-schoolgrades.However,theabstractionprocess in thisresearchiscarriedoutwithhigh-schoolstudentswithinanappropriatecontextbecausegivingknowledgedirectlytothestudentswhohavetroublecomprehendingsomethingstructurallyabstractdoesnotleadthemtobuildbuttomemorizethisknowledge.Thelimitproblemschosenbytheresearchersforthisstudyalsocontaindifferentcontexts,whichisvaluableandmeaningfultostudents.Accordingly,theresearchprobleminthisstudyhasbeenquantifiedas“Howdo12th-gradestudentsabstractlimitknowledge?”

Thisstudydiffersfromotherstudiesonthesubjectoflimitsinthatitallows12th-gradestudents’abstractionprocessforlimitstobeexaminedthroughconstructivistlearning, aswell as the learningprocesses in detail using theRBC+Cabstractionmodel.Moreover,thisstudyisconsideredtocontributetothefieldbyallowingforanunderstandingofstudents’insufficientknowledgeandmisconceptionsregardinglimitsandadeterminationofthedistinctwaysstudentsuseintheirlearningprocess.

MethodThis section includes information about the research model, the participating

students,andtheirabstractionprocesswhenlearningusingtheRBC+Cabstractionmodel.Inthisstudy,asignificantamountofdatahasbeenobtainedfromalimitednumber of students, and the abstraction process has been evaluated very clearly.Therefore,thisstudyisdefinedasacasestudy.

Case studiesoffer a convenientmodel for examiningahighlydetailed specificphenomenon, such as a person, a process, an operation, or an institution. In qualitative case studies, the researcher(s) explores a process, questions may address a case description, and the themes emerge from this (Creswell,2009,p.130). Additionally, choosing a single case can provide detailed and significant knowledge (Vural

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&Cenkseven, 2005). The case in this research is the detailed examination of the abstraction of limits. In case studies, obtained data is quite powerful in terms ofreality.Moreover,thefindingsareavailableforinterpretationanddirectuse(Cohen &Manion, 1994, p. 123).Additionally, when researchers are an observer in theresearch,theyarealsoabletobeaparticipantforevaluatingthesubjectasapartofthe process (Yıldırım&Şimşek,2005,pp.77–78). In this study, one of the researchers alsoparticipatedasanobserver/actor.Participantinterviewscanalsobeusefulforinterpretingmathematicalthoughtandabstractionduringtheinterview(Geray,2006,p. 171).As such, it is appropriate for the study andmakes itmore specifically adescriptivecasestudy,whichgenerallyrequiresobtainingdetailedinformationabouta phenomenon to ensure data for future comparative studies (Merriam,1988).

Sampling/Study PopulationThe participants of this studywere two 12th-grade students enrolled in an high

schoolinalargecityofTurkey’scentralAnatoliaregion.Bothstudentsarefemaleandtheirmathteachersreportedthemassuccessfulwithexcellentgrades in theirmathcourses. They are also very direct, curious math students. Additionally, these students voluntarily participated in the study and had not previously studied limits. They had alreadylearnedthesubjectsofseriesandfunctions,whichisneededtoacquirelimitknowledge.ThestudentsweregiventhemonikersofZehra(Z)andGizem(G).

Data Collection ToolsAbstractionhappensintheproblem-solvingprocess,butaccordingtoHershkowitz

etal.’s(2001)research,allproblemsdonotleadtoabstraction.Forthisreason,threeproblemswerearrangedbytheresearchersinordertoreachconstructionwithintheabstractionprocess,torevealstudents’mathematicalthoughtlevelsandknowledgeabstractionprocess,andtodevelopanewstructure.Theseproblemsareadditionallyfrom the areas students are quite likely to encounter in daily life, such as:

1.Aliheadstoatree20metersawaybyfirstcoveringhalfthedistance,thenhalfthe remaining distance, and again each time, half of the remaining distance. Will Alibeabletoreachthetreelikethis?

2.Twostudentsplayagameapproximatinganumberbyfollowingtherulesgivenbelow.Anumber is chosen.One student says two realnumbers,onealwaysbiggerandtheotheralwayssmaller thanthechosennumber.Thepurposeofthegameistosayanumberclosesttothechosennumber.Thestudentwhoisunabletosaynumberscloserthantheopponent’slosesthegame.Forexample,let’ssupposestudentsplaythegamechoosingthenumber“6.”

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1ststudent5.55.95.975.9925.9997……

2ndstudent6.56.096.026.0016.0003……

Whodoyouthinkwillwinthegame?

3.Ababykangaroois learningto jump.Everydayit trieshardto jumphigher.Letthenumberofjumpswhichthekangaroomakesbex. The height that this kangaroocanjumpisexpressedwiththefunctionoff(x) = (4x - 1) / 2x.

a)Calculatetheemptyboxesbelow.

xnumberofjumps125101001000100001000001000000+∞

f(x) in meters

b)Whatisthemaximumheightthekangaroocanjump?

c) If f: A→B,findsetsAandB.

d)Isthisfunctionasequenceatthesametime?Why?

e)Ifthisfunctionisasequence,whatvaluedoesitapproximate?

f) What is the desired x of function f(x) whengoingto+∞?

g)Istherearelationshipbetweenthedesiredvalueofafunctionandthevaluethatasequenceapproaches?

All theseproblemsrelatetothefirstandsecondacquisitionsofthehigh-schoolmathematics curriculum. Therefore, all these problems completely relate toabstractinglimitknowledge.Theaimofthefirstproblemistoletstudentsanticipatetheneedforlimitknowledgethroughtheirthoughtprocessandtoconstructthelimitfromonlyone side.The secondproblemaims togive students theopportunity tointuitivelyconstruct limitknowledgeapproachingapointand toconsolidate limitknowledgefromoneside.Additionally,thethirdoneisorganizedwithfunctionsforintuitively constructing limit knowledge as awhole.Accordingly, these problemsallow for constructing limitknowledgebyapproachingany realnumber from theright or left, and constructing a value upon a series that converges at the same time it is equal to the series limit. A research expert participated in the study in order to determinetheeaseofproblemsforapplyingtheRBC+Cmodel.Fiveexpertswerealsoasked for theiropinionsabout theeaseof theseproblems for learning limits,andtheproblemswererearrangedinthedirectionoftheirsuggestions.Forexample,“remaining”wasputinthefirstproblembytwooftheexpertssotheproblemcouldbesolvedcorrectly.

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ProcedureSemi-constructed interviewsandobservationswere thedatacollectionmethods

used in this research. Prior to the study, the aim and scope of the study wereexplained to the school administration and participating students in detail. Then in the fall semester of the 2014-2015 academic year, these students studied the research problemsatthesametimetogether.Withinthescopeofthisstudyandinordertoreachaconclusion,thestudentswereaskedtoapproachanyrealnumberfromtheright or left according to the information given for different events, form a list, and fillinthegapsinthegiventablebyusingthedatainthislist.Inthemeantime,oneoftheresearchersparticipatedinthestudyasaparticipantobserver.Theprocesswasrecordedwithavideocamerathatstudentscouldsee.Thiswastopreventforgettingtheenvironmentinwhichtheobservationsandinterviewswerebeingconducted.Atthesametime,therecordingsallowtheresearchertoreviewthemasoftenaswantedsoastodelayafinaldecisionuntilbeingsureaboutthemostappropriateapproachandwhichpointstohighlight.Hence,theresearcherwasabletofindrareorfrequentevents by fast-forwarding/rewinding the recording and, beforemaking a decisionabout an event, change or correct her interpretations by looking at the previous/subsequentevents(Plowman,1999,ascitedinTopbaş,2008).

Analysis of Research DataThe data was analyzed and interpreted using a descriptive analysis of the

worksheets, including theanswersstudentsgave to the researchproblemsand thevideosrecordedduringtheinterviews.Theaimofanalysisistopresenttheobtaineddatainanorganizedandinterpretedway.Withthisaim,theobtaineddataisdescribedsystematically and clearly (Yıldırım& Şimşek, 2005, p. 224). At this stage, the dialoguesrecordedduringtheinterviewwerelistenedtoseveraltimesandtranscribedinto statements/texts in the study. This data is to be examined, and the obtainedinformationshouldbedividedintomeaningfulsections(Kümbetoğlu,2005,p.155). Accordingly, theRBC+Cmodelwas applied to the statements in the application/abstractionprocess,andthesestatementswereanalyzedoverthecognitiveactionsofthisabstractionmodel.Inthiscontext,thethemesdeterminedandusedforanalyzingthewritten interview statements are recognizing, building-with, constructing, andconsolidating. Inotherwords, thesecognitiveactionswereobservedandrecordedcollectivelyinthesolutionofeachproblembytwooftheresearchers,independentofeachother.ThedefinitionsofthesecognitiveactionsandexamplesrelatedtotheseactionsfortheproblemsinthisresearcharegiveninTable1.

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Table1Cognitive Actions of the RBC+C Abstraction ModelAction Definition Examples

Recognizing

Includes formal or informal information thatanindividualhasacquiredbeforeandtheirattributionofmeaningtomathemati-cal elements in the learning environment. Itraisesindividualawarenessofwhatkindof foreknowledge is needed in order tolearnnewinformation.

*Recognizingtousesequenceinthefirstproblem.*Recognizingtousefunctioninthefirstproblem.* Recognizing infinity in the second researchproblem.*Recognizinglimitknowledgefromonesideinthethirdproblem.

Building-with

Relates to individualuseof thepiecesofinformationthatonealreadyknowstocon-structasolutionapplicabletotheproblem.This cognitive action is nested in the other.

*Usingsequenceandseriesknowledgetosolvethefirstproblem.*Using infinityknowledge to solve thesecondproblem.* Using limit knowledge from one side in thethirdproblem* Using functions to solve the first and thirdproblem.

Construction Ispartlychangingknownstructuresand,asaresultofthis,constructinganewstructure.

*Constructinglimitknowledgefromonesidebyrecognizingandbuilding-withthesequence,se-ries,andfunctionsinthefirstproblem.*Constructinglimitknowledgeatapointinthethirdproblem.

ConsolidationRequires enhancing the new structure(Dreyfusetal.,2001a;Hershkowitzetal.,2001).

* Consolidating limit from the left or limit from therightinthesecondproblem.

Inthiscontext,studentsrecognizetheirforeknowledge,build-withthisknowledgefor different purposes, and examine thinking styles used during the process of limit-knowledge construction.Then the extent that students have constructed their limitknowledgeandthepre-structuresthatwererequiredisreported.Finally,interpretationsweregivenaccordingtothedatainordertomakesenseofthedescribedfindings.Therelationshipsbetweenthesefindingsandtheconclusionsdrawnarepresentedindetail.

Validity and Reliability of the StudyThevalidityandreliabilityofthisqualitativeresearchstudyhavebeenachieved

by taking the strategies of conclusiveness, transferability, and reliability intoconsideration (Yıldırım&Şimşek,2005,pp.264–272).

In this study, conclusiveness and consistency were achieved through expertexamination.Firstanexpertworking in thefieldwasasked todetermine theeaseof the problems, and five different experts were asked their opinions about howlimitknowledgeislearnedthroughtheseproblems.Afterthat,theseproblemswererearrangedpertheirsuggestions.Additionally,thevideorecordingswereexaminedandtransferredtotextsindependentlybyeachofthetworesearchersandplacedintothedeterminedthemes.Atthisstage,thethemeswereclassifiedintogroupspreparedbytheresearchers,andtheresearchersresolvedanydiscrepanciesthroughdiscussionand evaluation. While describing and interpreting the typical and changeable

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characteristicsofthestudyindetailwereusedtoachieve transferability,quotingfromtheobtaineddataasmuchaspossibletoobtainsufficientdataforanalysiswasusedtoachievereliability.Additionally,importancewasgiventoclassifyingresearchdatainaccordancewiththedescriptivemethod,aswellastoorganizingandinterpretingthedeterminedthemes.Moreover,attentionwaspaidtowritingtheresearchfindingswiththecharacteristicsofreasonability,appropriatenessforindividualexperiences,conclusiveness,importance,andreadability.

FindingsThe interview was carried out over two successful 12th-grade students who

participatedinthestudy.Theseparticipantsstudiedthreedifferentproblemsinthisprocess,whichlastedapproximately45minutes.

Thefirstproblem(Aliheads toa tree20metersawaybyfirstcoveringhalf thedistance, then half the remaining distance, and again each time, half of the remaining distance.WillAlibeabletoreachthetreelikethis?)waspreparedwiththeaimofhavingthestudentsanticipatetheneedforlimitknowledgeusingtheirthoughtprocessfrom only one side of the situation. At this stage, the students need to estimate each step of the procession. In this process, it is necessary to recognize andbuild-with,twocognitiveactionsrelatedtotheRBC+Cabstractionmodel,andknowingsequenceandseriesarethepre-knowledgeusedinsolvingtheproblem.Therefore,thesestudentsareexpectedtothinkaboutthesituationandinferencelimitknowledgeaccordingly.

The students read this problem together and started to discuss it. Below is anexcerpt of their discussion.

Zehra5:Gonehalfatatime.(Looksatthepaperandthinks)

Gizem6:Graduallydecreasing…

Z7:(Nodsherhead.)

G8:Triestoapproximatebygraduallydecreasing.

Z9:Yes…(Nodsherheadagainandlooksatthepaper)

G10:Thesearethesubjectsthatwesawlastyear,Imean;whenwewerein11thgrade…(Shepointstotheproblemwithherpencilandlooksatherfriend)

Z11:Series…(Looksatherfriend)

G12:(Looksatherfriendandnodsherhead)Actually,itcanbedonebyreasoningtheseries…

Z13:Byusingthetotalsymbol…(Playswithherearandthinks)

G14:Byusingthetotalproductsymbol…(Readstheproblembyunderliningthesentenceoftheproblemwithherpencil.)Sheasks,“CanAlireachthetree?”IthinkAlican.(Zehraapprovesherfriend’sopinionbynoddingherhead)

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Z15:Allinall,20metersisalimiteddistance.Sincethedistanceisinmeters,itcanbecovered.

G16:(Nodsherheadinapprovalofwhatherfriendhassaid)Thedistanceisinmeters,itcanbecovered.

Z17:Canbecovered,ImeanIguessitcanbestepped.

G18:(Zehrapointstoherfriend’sworksheet)1.25,aswedoinseries,wellthis…

Z19:Well...hmm…(Triestoremember;tapsthesheetwithherpencilandhands)

G20:Hmm...Ifwewrite20intheformulaoftotalfunction…

Z21:Wewrite20.

G22:Hmm…

Z23:Willwestartfrombelow1,too?(Pointstotheproblemonherfriend’ssheet)Won’twestartfrom1?Startingfrom1until20…(Looksatherfriendasifwaitingforapproval)

G24:Isit1?How?

Z25:Well,intheend,healwayscovershalfthedistance…

G26:Hegoesinseries.

Z27:Hewentinseries.

(They talk at the same time looking at each other and nodding their heads in approval).

G28:Half,half,half,half.

Z29:Half,half,half.

Researcher30:HowdoyouthinkAlicanreachthetreemathematically?(Inthemeantime,thestudentslookattheresearchercarefully)Canyourelateittothesubjectofseriesthatwesawin11thgrade?

G31:Ithinkhecan…(Firstshelooksatthesheet,thanexplains)Inthelesson,wedroppedtheballfromahighplace.Thenitreachedthehighest[bounce],thisisthesame.(Shelooksattheresearcherandherfriendandshowstheball’spathofmovementbouncingwithdecreasingheightusingherhand)Justatreeintervenes.It’sthesamehere,too.Acertain…???one.Atacertainstep.

Z32:(Pointsattheproblem)Imean,intheend,anet...betweenAliandthetree…

G33:Distance…

Z34:Hegraduallyapproximatessinceitisadistance.

G35:Yes. (They nod their heads, looking at each other.) He will go at a certain function,decreasing,butintheend,(markedtheplacewherethetreeis)hereaches.

Z36:Hewillreach.

G37:Thereasoningisthesameinseries…(Pointstotheproblemtextinthesheetwithherfinger)Ifweputitinseries,wecansolveitovertheseries.

R38:Well,whatkindofsequenceisit?Isitageometricalsequenceoranarithmeticsequence?

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G39:Ageometricalsequence(Looksatherfriend’sfaceandwaitsforapproval)

Z40:Ageometricalsequence…(Looksattheresearcher)

R41:Ageometricalsequence?Canyouobtainanetresultwiththeformulaofageometricalsequence?

G42:Yes…(Thestudentsnodtheirheads)

Z43:Remember,thehighestandthesmallestnumber…

R44:Doyourememberhowyoudidit?

G45:Rememberwhat?Theformula?

R46:If suchaproblemwereasked in thesubjectof sequences,doyou rememberhowyouwouldsolveit?(Theylookatthesheet).

G47:Wewritethesummationsymbol,thenwewrite20ontopsideandstartfrom0anddecreaseby(½)n.Weputitinthebracketof½,½plussquareof½plus(½)n…(Pointedtotheworksheetwithpencil)Intheend,wewillequalizeitto20andfromthatsequence…well,we’llbeabletofindthatsequence.(Inthemeantime,Zehralooksattheresearcher,herfriend,andtheworksheet.)

R48:Youweresayingthathewouldreachthistreeintheend?

G49:Yes,hereachesthereintheend.

R50:Bydoingproceduresinthisway.(Thestudentsnodtheirheads.)

Theinterviewtextsgivenhere(Z11,G12,Z13,G14,G18,G20,G26,Z27,G35,Z36,G37,G39,Z43,G47andG49)indicatethatthestudentsrecognizedknowingthesequence, series,and functionarenecessary tosolve theproblemand thoughttobuild-withinthesolutionofproblem.TheseinterviewtextsindicatethefactthatGizemandZehraconstructedthelimitfromoneside.

Afterthat,thesecondproblemwasdirectedtothestudentsasfollows:

Twostudentsplayagameapproachinganumberbyfollowingtherulesgivenbelow.Anumberischosen.Onestudentsaystworealnumbers,onealwaysgreaterandtheotheralwayslessthanthechosennumber.Thepurposeof thegameis tosaynumbers thatarecloser to thechosennumberthanthepreviousones.Thestudentwhoisunabletosayanumbercloserthanthisrival’slosesthegame.Forexample,let’ssupposethestudentsplaythegameforthenumber“6.”

1ststudent5.55.95.975.9925.9997……

2ndstudent6.56.096.026.0016.0003……

Whichonedoyouthinkwillwinthegame?

Thissecondproblemwaspreparedwiththeaimofhavingthestudentsrecognizeandbuild-withinfinityaspre-knowledgeinsolvingtheproblem.Additionally,they


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consolidatedknowingthelimitfromoneside,andbegantorecognizeandbuild-withittoconstructknowledgeofthelimitatapoint.Inotherwords,studentswereexpectedtoknowhowtoconstructthelimitatapointinthisproblem.Atthisstage,theyreadtheproblemandthendiscusseditforawhile.Theconversationtextisgivenbelow.

Z51:Nowwehavethenumber‘6’athand.Onesaysanumbergreaterthan6,theothersaysanumberlessthan6.

G52:(Shenodsherheadinapprovalofwhatherfriendissaying)Andtheothersaysanumberless than 6.

Z53:Asaresult,always…(Shedrawsherhandsnearandaway)

G54:Move themaway fromeachother and say things.Well… (Shows the numbers in thesheet)Itseemsthat5.9islessthan6.001.Onesaysnumbersthatgetsmaller,theothersaysnumbersgreaterthan6.(Zehranodsherheadinapprovalofherfriend)Butwhichonewins?(Theylookatthesheetandthinkitover.

Z55:Asaresult,oneisdecreasingwhiletheotherisincreasing…

G56:Ithink…no…(Sheshakesherheadindisapproval)

Z57:Ineveryway,thesame...(Shedrawsherhandsnear)

G58:Theysaythesame.Theycanalwayssaynumberslessthan6.Thosewhicharegreaterthan5aswell.

Z59:Thenumberswhicharegreaterthan6aswell…Sheaskswhichstudentwins…Whichonesaysaclosernumber?(Looksatherfriend.Gizemalsopointstothenumbersonthesheet;Zehralooksather.)

G60:Now,Iwonder,whenthissays6.02,willtheothersay5.9again?(Zehrapointsatthesheet)

Z61:Theysaidthese.Whenonesaid6.02,theothersaid5.97.

G62:Hewillneversaythosenumbersagain...(Zehraagreeswithherfriend)Thennumbersgreaterthan6cangooninfinitely.Imeanhehasachancetosayallthenumbers.

Z63:Yes,Imean,itgoesonineveryway.

G64:Andthistothelesserones…(Shelooksatherfriendasifwaitingforapproval)

Z65:Hewillsaythesmallestnumbersinfinitely.Imeantheywillnotcomeacross.

G66:Ithinktheywilltie.(Zehranodsherheadagreeingwithherfriend.Thentheylookattheresearcher.)

Z67:Noonewillwin.

R68:Whydidyoudrawsuchaconclusion?

Z69:Becausethereareinfinitelymanynumbersonbothsides…Bothlessthanandgreaterthan6.

R70:Wecanchooseinfinitelymanynumbersthatarecloserto6.(Thestudentsnodtheirheadsinagreementwiththeresearcher).

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Theinterviewtextsgivenhere(G62,Z63,G64,Z65,andZ69)indicatethatthestudentsrecognizedtheconceptofinfinity,whichisnecessarywhensolvingforit,andbuilt-withitinsolvingtheproblem.Accordingly,thetexts(Z51,G52,Z55,andG58)givenhereshowthatZehraandGizemconsolidatedthelimitofoneside.Othertexts(G60,Z61,G62,andZ63)showthattheybegantorecognizeandbuild-withthisknowledge.Additionally,some(Z57,G58,G62,Z63,G64,G66,Z67,andZ69)ledresearcherstothinkthestudentsbeganatonepointtointuitivelyconstructlimitknowledge.

Finally,thefollowingwasgiventothestudentsasthethirdresearchproblem.

Ababykangarooislearningtojump.Everydayittrieshardtojumphigher.Letthenumberofjumpsthekangaroomakesbex.Theheightthatthiskangaroocanjumpisexpressedwiththefunction of f(x) = (4x-1)/2x.

Thislastproblemwasorganizedwithfunctionstoformallyconstructstudents’limitknowledgeasawhole.Theywereexpectedtorecognizeandbuild-withknowledgesuchasfunction,infinity,realnumbers,andsequenceaspre-knowledgeinsolvingthisproblem.Thestudentsneedtorecognizeandbuild-withtheirknowledgeoflimitfromtheleftandlimitfromright,andtoconstructknowledgeoflimitatapointinthisproblem.Atthisstage,theyreadtheproblemandthendiscusseditforawhile.Theconversationtextisgivenbelow.

Firstofall,thestudentswereaskedtoreadSectionAoftheproblem:

Calculatetheemptyboxesbelow.

Xnumberofjumps125101001000100001000001000000+∞

f(x) in meters

Thentheywereaskedtointerpretittogetherandfillinthespacesleftblankinthetable.Apartialtextbelongingtotheinterviewheldatthistimeisgivenbelow.

Z71:Whenxis1,itbecomes3/2.

G72:Whenweput2,weget7/4.(Shelooksatherfriendasifwaitingforapproval)Whenweput5,weget19/10.Whenweput10,weget39/20...Itgoeslikethis.(Zehraapprovesandwrites,too.)

G73:Let’stry1000.3,999/2,000.Whenweput10,000,weget399,999/200,000.(Theywritetheir results).

The partial interview text given here (Z71, G72, and G73) indicates that thestudentsrecognizedandbuilt-withtheknowledgeoffunction,whichisnecessaryforsolvingtheproblem.Thenthestudentscontinuedtowork,answeringSectionBoftheproblem(Whatisthemaximumheightinmetersthatthekangaroocanjump?).

G76:Itisinmeters.Graduallydecreasingfrom3/2…Imeanasthenumberofjumpsincreases.(Zehranodsherhead)

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Z77:Yes,thenthemaximumvaluewillbeitsfirstvalue…

G78:Itsmaximumvalueisitsfirstvalue…Thehighestvaluebecomes3/2,or1.5meters...

Z79:(Nodsherhead)1.5meters.Asthenumberofjumpsincreases…

G80:For as the number of jumps, its value decreases in meters. For this reason, 3/2…maximum…

R81:Canyouexplaininmoredetail?

G82:Asthenumberofjumpsincreases,f(x) gets smaller.

Z83:Asiff(x) goesinfinitelybygettingsmaller…

R84:Examineoncemore,ifyoulike.(Thestudentslookatthesheet)

G85:No,bigger.1.75looks…

Z86:Oh!Right…(Gizemnodsherheadandpointstothetablewithherpencil)

G87:Asxnumberofjumpsincreases,sodoesthevalueofthef(x)function.Imeanwhenx jumps once(1),itbecomes1.5meters,butwhenitjumpstwice(2),itbecomes7/9,Imeanlike1.75…

Z88:Lateronitgoesto1.9,anyway.

G89:Yes,afterthat1.95.Imeanasxnumberofjumpsincreases,thevalueofthefunctiongetsbigger,too.But,wecannotknowitsmaximumvalue.(Theylookateachother)Oneminute,asifitgraduallyapproximatesto2.(Zehraputsherhandonherchin)

R90:Whichvaluesdidyoufind?

G91:Firstwe found1.5, then 1.75, 1.9, 1.95, and 1.9995. Finally, it approximates to 2. Itapproximates to a maximum of 2.

Z92:(ZehralooksatGizemandnodsherhead)Actuallyitapproaches2,thevaluesaresmallerthan2.

G93:Yes.Wecannotfinditshighestvalue.

Some interview texts given here (G85,Z86,G87,Z88,G89,G91,Z92, andG93)indicatethatthestudentsrecognizedandbuilt-withtheknowledgeofinfinity,functionandthelimitfromoneside,whichisnecessaryforsolvingtheproblem.Afterthat,thestudentswereaskedtoanswerSectionCoftheproblem(Iff: A→B,findsetsAandB).

Z99:Then,willwethinklikeAgoingfrom1.5to2?

G100:Thedefinitionset.(Gizemmarksthecolumnwherexiswithherpencilandshowsittoher friend).

R101:Yes,asthedefinitionsetandtheimageset…

Z102:Ok,Ok…(Zehranodsherhead)

G103:x is thedefinitionset,andthevalueswhichwefoundinmeters type(sheshowsherfriendthef(x)columnthatshefoundinthetable)willcomposethevalueset.Whenwewrite1,itsvaluebecomes1.5.Then,whenwewrite2,itbecomes1.75.Thedefinitionsetisxnumberofjumps,andtheimagesetisitsvalueinmeterskind.(Zehraapprovedbynoddingherhead)

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R104:Inyouropinion,whatisthedefinitionset?Whenwethinkofsets?Integer?Rationalnumber?

Z105:Thedefinitionsethastobeaninteger.

R106:How?

G107:Positiveintegers.

Z108:Sincexindicatesthenumberofjumps,x(numberofjumps)cannotbenegative…

G109:Ithastobeapositiveinteger.

Z110:ImeanIcannotsayitisarealnumber.

R111:Well,inyouropinion,whichvaluesinwhichintervaldoesthevaluesettake?

Z112:Theintervalof1.5-2…

G113:Thevaluesapproaching1.5and2.Not2but…(Zehraapprovesbynoddingherhead)

Fromsomeoftheinterviewtextgivenhere(Z105,G107,G109,andZ110),thestudents are understood to have had difficulty recognizing and building-with theknowledgeofrealnumbersnecessaryfordeterminingtheintervalsinthispartofthequestionwhilesolvingtheproblem,buttheydidmanageit.ThestudentsthenheldthefollowingconversationwiththeaimofansweringSectionDoftheproblem(Isthisfunctionasequenceatthesametime?Why?).

Z115:Now,accordingtothesequenceswhichwelearnedin11thgrade,itincreasedbygoingtoinfinitybutproportionally…

G116:Wedonotknowitsvaluecompletely.Weknowwhichnumberitapproaches.Wedonotknowthefullvalue.

Z117:Imean,yes,whenxwenttoinfinity,itincreasedinthesameway.

G118:Remember,thereweresequenceproblems.Forthatreason,thereisthesamethinginthisquestion,too,itapproachesavalue,but…

Z119:Yes.

R120:Well,canyoudefineasequenceforme?Doyourememberwhatasequencewas?Whatisneededforafunctiontobeasequence?

G121:Firstofall…(Looksatthesheet)

Z122:(Lookingatherfriend)Whenanumbergoestoinfinity,ityieldsthevalueitwilltake.

G123:Itneedstoincreaseordecreaseproportionally.(Inthemeantime,Zehranodsherheadandagreeswithherfriend)Whenitincreasesby1.5,theothersneedtoincreaseby1.5,too.

R124:Canitbethedefinitionofanarithmeticsequence,whatyousaid?Doyourememberhowyoudefinedthesequence?

G125:...whichpositiveintegerstake…(Sheputsherhandonherchin)

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Z126:Thevalueswhichpositiveintegerstakewhengoingtoinfinity.

R127:Well,doyouthinkourjumpfunctionisasequence,too?

Z128:Yes,itisanintegergoingtoinfinity.

G129:Ithinkitisasequence.

Z130:Itisasequence.

Theinterviewtextsgivenaboveindicatethatthestudentsrecognizedandbuilt-withtheirknowledgeofsequence,whichisalsonecessarytoanswerthequestionwhilesolvingtheproblem.ThenthestudentshadthefollowingconversationinordertoanswerSectionEoftheproblem(Ifthisfunctionisasequence,whatvaluedoesitapproach?)

R131:Well,ifthisfunctionisasequence,whatvaluedoyouthinkitapproaches?Whichvaluedoesthisfunctiongoto?

G132:Itgoestothevalueof2.Forexample;1.9999…

R133:Howdidyouarriveatthisconclusion?

G134:Whenwereplacethevaluesinthedefinitionsetwiththefunction,thevaluesincrease.Westartedwith1.5andIthinksinceitapproached1.99995,itcanatmostapproach2.

ThenthestudentsheldthefollowingconversationinordertoanswerSectionFoftheproblem(Whatdoyouthinkisthedesiredxvalueoffunctionf(x)whenitgoesto+∞?)

G138:Wewanted to reach2 in this sequence.For this reason, it is 2. (She shows the f(x)columnwithherfinger.)

Z139:Weapproach2andthenumbersarelessthan2.

G140:From the left.Weapproach from the lefton thenumber line. (Shebringsherhandstogetherasifshowingthenumberline)

Z141:Itseemsasiftapproaches2fromtheleft.(Shebringsherhandstogetherasifshowingthenumberline)

G142:Weapproachitfromleft.(Zehranodsherhead)Ithinkthedesiredvalueofthesequenceis2.

Withtheseresearchproblems,thestudentsareexpectedtogeneralizetheresultsandmakerelationsbetweenthefunctionandsequencetointuitivelyconstructtheknowledgeofthelimitatapoint.Theinterviewtextsgivenhere(G138,Z139,G140,Z141,andG142)letoneconsiderthatthestudentsconstructedtheknowledgeofthelimitbybuilding-withthelimitfromoneside.Finally,theconversationheldbythestudentsforansweringthequestionoftheproblem(Istherearelationshipbetweenthedesiredpointofafunctionandthevaluetowhichasequenceapproaches?Pleasediscuss)isgivenbelow.

Z145:Nowthisquestionhasasequence.Asequencecomposedofintegers.Itcomesfromtheleftside.Andourdesiredvalueis2andweapproachthisnumberwithnumberslessthan2.Nowthattherewillbeanintersectionpointbetweentwointheend,thisaccumulationpointmaybeanadditionaldesiredpointofthesequence.

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G146:Onemore thing, rememberwhatwe said...We are trying to approach itwith lessernumbers.Herewearetryingtoapproach2.Imeantherecanbearelationshipbetweenthe value we are trying to approach in the sequence and the desired value. (Zehraapprovesbynoddingherhead)Then thedesiredvalue is related to the smallorbignumbersinthesequence.

Z147:Itisadvancingbyincreasingordecreasing.

G148:Forexample,thereweresuchkindsofquestionsinthesequence,too.Aballwasdroppedfromaheightanditsbouncingheightdecreased…(Sheshowedwithherhand)Andfromhere,weareapproachingfromasmallnumber toagreaternumber…But theresultwhichwewanttoreachisthesame.

Z149:Yes,definitely.

G150:Ifithadstartedwith1,000bounces,itwouldstillhavereached2.Butitstartedwith1bounce.

Z151:Asaresult,itagainreached2withlessernumbers.Andagainitsdesiredvalueis...

G152:Itsdesiredvalueis2.Imean,thereisarelationshipbetweenthem.

R153:Well,Whatdidyoucalculateasthelimitwhenitgoestoplusinfinityinthisexample?

G154:2(Zehranodsherhead).

R155:Yousaid2. Ifour f(x) function isasequence,whatdidyoucalculateas thevalue itapproaches?

G156:Wecalculateditas2.

R157:Consideringthis,whatcanyousay?Whatkindofaninterpretationappears?

Z158:Imean…asaresultitisanarithmeticsequence…

G159:Doeseverysequencehaveadesiredvalue?

Z160:Everysequencehasadesiredvalue.

G161:Becauseboththesequenceandthedesiredvaluearetryingtoapproach.

Z162:Approachordeviate.

G163:Inmyopinion,asequence…(Thinksforafewseconds)Asifthelimitistheupperstageof the sequence.

Z164:Ithinkthedesiredvaluecoversthesequence…

G165:Yes,I’mthinkingthesamething.

R166:Youfoundthedesiredvalueofthefunctionas2.Andyoufoundthevaluetowhichthesequenceapproachesas2,also.Then,whenitgoestoinfinity?

Z167:Thedesiredvalueofasequencegoingtoinfinity…

G168:Isitthesame?Theyareequal.

Z169:Yes.

R170:Whatdoyouthinkareequal?Interpret.(Theylookateachotherandthinkforafewseconds)

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G171:Both…

Z172:Asaresult,itgoestoinfinity.Italsogoestoinfinitywhenittakesthedesiredvalue.

G173:Asaresult,itisafullvalue,asshefound,butthesequenceandthedesiredvaluearethesame.Then,whenthefunctionf(x)approaches2,thisapproaches2,aswell.Thenthesequenceandthedesiredvaluearethesame.(Sheshowsthetable)

Z174:Becausethevalueitapproachesis2.Actually,theyareequalininfinity.Whenitgoestoinfinity…

G175:Itagaintakesthesamevalue.

Z176:Thevaluetowhichanumber,asequence,approximatesandthedesiredvalueisequalforinfinity.(Theynodtheirheads)

R177:Yes,andthisdesiredvalueiscalledthelimitvalue.

Someoftheinterviewtextsgivenhere(Z145,G146,G148,Z149,andZ151)pointoutthatthestudentsrecognizedandbuilt-withthelimitfromtheleftsidetoconstructlimitknowledge.Similarly,othertexts(G146,Z147,G148,Z149,G152,G156,Z160,Z167,G168,Z172,G173,Z174,G175,andZ176)showtheyrecognizedandbuilt-withtheconceptsofsequence,function,andinfinitytoconstructthelimitatapoint.

DiscussionTheabstractionprocessesofthetwo12th-gradestudentswereexaminedthroughthe

RBC+Cabstractionmodel in this research.The students’ processeswere examinedusing three research problems over the cognitive actions of recognizing, building-with, construction, and consolidation. These problems, which were designed bythe researchers to reveal the students’ mathematical thought-levels and abstractionprocesses,enabledthestudentstousetheirpre-knowledgeandachieveanewstructure.

Recognizing,thefirstcognitiveactionofthismodel,isnecessarytoconstructthenewstructure.Also,thisactionrequiresusingthesecondaction,whichmeansrecognizingandbuilding-withneedtobeusedtogethertosolvetheproblems.Thisisbecausethefirstonecontainsrecognizingtheconceptofpre-foreknowledgetomakeiteasierfortheparticipants to study theproblem in theprocessof construction and the secondone requires them to use their pre-knowledge for this construction (Dreyfus, 2007;Hershkowitzetal.,2001). For this study, the students used the concepts of sequence, series, infinity, realnumbers,andfunctionaspre-knowledge toconstruct limits.Allof theseconceptswere learned in theirmathematicscoursesprior to thisstudy.Forexample, real numbers occur in themathematics curriculum as a 9th-grade subject,and functions occur as 9th- and 12th-grade subjects. In the12th-grade, knowledgeoffunctionsisthesubjectlearnedbeforelimits.Therefore,thestudentswereunderstoodtorecognizeandbuild-withtheconceptsofsequence,series,andfunctioninthelearningprocessof limitknowledgeaspre-knowledge for solving thefirst limitproblem,as

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seeninconversationtextssuchasZ11,G14,Z27,G35,Z43andG47fortheabstractionprocessesanalyzedthroughthecognitiveactionsoftheRBC+Cabstractionmodelinthis research.Similarly, they recognizedandbuilt-with theknowledgeof infinity inthe learning process of limit knowledge for the secondproblem.Additionally, theybegantorecognizeandbuild-withknowledgeofapproachingthelimitfromonesidein thisproblem.These female students recognizedandbuilt-with theknowledgeofinfinity,function,andlimitfromonesidetoconstructtheirlimitknowledgeatapointin the thirdproblem.Theyalsohadsomedifficulties recognizingandbuilding-withknowledgeofrealnumbers,butwereabletomanagethis,aswell.Theyrecognizedandbuilt-withknowledgeofapproachingalimitfromonesideandofsequence,function,and infinity for constructing the limit at apoint, as seen such in textsG146,Z172,G173,Z174,G175,andZ176.Thissituationindicatesthattheparticipatingstudentscouldexerttheirpre-knowledgeforlimitsandexplaintherelationshipsbetweenthem.

Constructingisthemostimportantactintheabstractionprocessbecauseitrequiresa new structure to emerge using pre-knowledge, or in otherwords, changing knownstructures,recognizingandbuilding-withinthelearningprocess(Dreyfus et al., 2001a; Hershkowitzetal.,2001).Inthisaction,participantsneedtoorganizeandcombinealltheirpre-knowledgetofindasolution.Inthisresearch,aconstructivelearningenvironmentwasorganizedforabstractingstudents’limitknowledge,andasaresultoftheresearch,theywereunderstoodtohaveconstructedthelimitfromonesideinthesolutionofthefirstproblem,asseenintextssuchasZ13,G39,Z43,G47,andG49.Similarly,theybegantoconstructthelimitknowledgeatapoint,asseenintextssuchasZ57,G58,Z67,andZ69whensolvingthesecondproblem,andconstructeditinthethirdproblem.

Thelastaction,consolidation,wasaddedtothisabstractionmodelanddetailedby Dreyfus(2007).ThemodelcanbesaidtobeclarifiedandfinalizedinDreyfus’research, and after that, researchers (Altun&Yılmaz,2008;Sezgin-Memnun&Altun,2012;Yeşildere&Türnüklü,2008a,2008b;etc.) generally focused on applying this modelusingdifferentmathematicalsubjects.Inthisresearch,someoftheinterviewtextsindicatethatthestudentsconsolidatedknowledgeofthelimitfromonesideinthesolutionofthesecondproblem.

TheRBC+Cabstractionmodelexplainstheroutethatleadstoabstractknowledge,asopposedtojustlearningit.Asaresultoftheanalysismadeinrelationtothisabstractionmodel’scognitiveactionsofrecognizing,building-with,constructionandconsolidation,ZehraandGizem,whosemathematicalachievementlevelswerehigh,canbesaidtohaveconstructedlimitknowledge.Thissituationindicatesdevelopingstudents’mentalstructures confirms these cognitive actions. Also, these explanations identify thatabstractionwascompletedbythesefemalestudents.Otherstudies(Altun&Yılmaz,2008;Sezgin-Memnun&Altun,2012;Yeşildere&Türnüklü,2008a;etc.) have focused

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on the abstraction of differentmathematical subjects, especially inTurkey, but theygenerallyreport thatstudentsconstructtheconceptbeingresearched.Inthisprocess,theyrecognizedandbuilt-withtheknowledgeofsequence,series,infinity,realnumbersand function.Moreover, the students in this study were also observed to not havemisconceptionsaboutlimits,indicatingthatthiskindofstudydemonstratesaneffectivelearningprocessforlimitknowledge.ThisresultoverlapswiththeresultsobtainedbyAkkoyunluetal.(2003)andBukova-Güzel(2007)in that it is appropriate to make use ofactivities,scenarios,subject-specificteachingstrategies,andworksheetsforteachinglimits.Additionally, these resultsare similar to thoseobtainedbyKula and Bukova-Güzel(2015b)becausetheyexplainthatpre-serviceteachers’specialteachingmethodsareclassifiedasrepresentationsandactivities.Atthesametime,thisstudystartedfromvarious small cases, and no misconceptions occurred. This result again supports the resultsobtainedbyAkkoyunluetal.(2003),inthatmisconceptionscanalsobepreventedinlessonsbyfocusingonexaminingthebehavioroftheimageofafunctionwhentheindependentvariableofthisfunctionisinfinitelysmall.Theparticipantswerefemalestudents in this studyandwereable toconstruct the limits in their learningprocess,whichwereorganizedwithproblem-solvingactivitiesthroughconstructivistlearning,similartotheresearchobtainedbyRoh(2007)withafemalestudent.

Limitations and SuggestionsThis research is limited by the two participating 12th-grade students and three

researchproblemsorganizedbytheresearchersforthisstudy.Therefore,researcherscouldexaminetheabstractionprocessofstudentsfromdifferentgradesoverdifferentmathematical subjects using the RBC+C abstraction model in future studies.Additionally, further research might also include course designs that allow forteachingdifferentsubjectsinaccordancewiththeRBC+Cabstractionmodel.

ReferencesAkbulut,K.,&Işık,A.(2005).Limitkavramınınanlaşılmasındaetkileşimliöğretimstratejisinin

etkinliğinin incelenmesi ve bu süreçte karşılaşılan kavram yanılgıları [Investigation ofeffectiveness of interactive teaching strategy on understanding of limit concept and encountered misconceptions in this process]. Kastamonu Education Journal, 13(2),497–512.

Akkoyunlu, A., Güler, M., Uğurel, I., & Alan, E. (2003). Ortaöğretimde limit kavramının oluşturulmasına yönelik bir çalışma [A research on the construction of limit concept in high schools].Retrievedfromwww.matder.org.tr

Altun, M. (2014). Liselerde matematik öğretimi [Mathematics teaching in high schools]. Bursa, Turkey: Aktüel.

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The Abstraction Process of Limit Knowledge · 2017. 10. 17. · 347 Sezgin Memnun, Aydın,...

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