startsstudentmain

Publishedonline:26December2019InternationalJournalofScienceandMathematicsEducation(2020)18:1673–1694https://doi.
org/10.
1007/s10763-019-10039-8*HugoBronkhorsth.
bronkhorst@rug.
nl1InstituteforScienceEducationandCommunication,FacultyofScienceandEngineering,UniversityofGroningen,P.
O.
Box407,9700AKGroningen,TheNetherlands2TeacherEducation,FacultyofBehaviouralandSocialSciences,UniversityofGroningen,GroteKruisstraat2/1,9712TSGroningen,TheNetherlandsLogicalReasoninginFormalandEverydayReasoningTasksHugoBronkhorst1&GerritRoorda2&CorSuhre2&MartinGoedhart1Received:7February2019/Accepted:1December2019/#TheAuthor(s)2019AbstractLogicalreasoningisofgreatsocietalimportanceand,asstressedbythetwenty-firstcenturyskillsframework,alsoseenasakeyaspectforthedevelopmentofcriticalthinking.
Thisstudyaimsatexploringsecondaryschoolstudents'logicalreasoningstrategiesinformalreasoningandeverydayreasoningtasks.
Withtask-basedinterviewsamong416-and17-year-oldpre-universitystudents,weexploredtheirreasoningstrategiesandthereasoningdifficultiestheyencounter.
Inthisarticle,wepresentresultsfromlinearorderingtasks,taskswithinvalidsyllogismsandataskwithimplicitreasoninginanewspaperarticle.
Thelinearorderingtasksandthetaskswithinvalidsyllogismsarepresentedformally(withsymbols)andnon-formallyinordinarylan-guage(withoutsymbols).
Intasksthatwerefamiliartoourstudents,theyusedrule-basedreasoningstrategiesandprovidedcorrectanswersalthoughtheirinitialinterpre-tationdiffered.
Intasksthatwereunfamiliartoourstudents,theyalmostalwaysusedinformalinterpretationsandtheiranswerswereinfluencedbytheirownknowledge.
Whenworkingonthenewspaperarticletask,thestudentsdidnotusestrongformalschemes,whichcouldhaveprovidedaclearoverview.
Attheendofthearticle,wepresentaschemeshowingwhichreasoningstrategiesareusedbystudentsindifferenttypesoftasks.
Thisschememightincreaseteachers'awarenessofthevarietyinreasoningstrategiesandcanguideclassroomdiscourseduringcoursesonlogicalreasoning.
Wesuggestthatusingsuitableformalisationsandvisualisationsmightstructureandimprovestudents'reasoningaswell.
KeywordsTwenty-firstcenturyskills.
Criticalthinking.
Formalandeverydayreasoningtasks.
Logicalreasoning.
SecondarymathematicseducationIntroductionP21'sFrameworkfortwenty-firstCenturyLearningdescribescriticalthinkingasanimportantskilltobesuccessfulinprofessionalandeverydaylifesituationsinanincreasinglycomplexworld(P21,2015).
Ofgreatvalueforcriticalthinkingis'reasoneffectively',whichisexplainedinthetwenty-firstcenturyskillsframeworkas"[using]varioustypesofreasoning(inductive,deductive,etc.
)asappropriatetothesituation"(P21,2015,p.
4).
Liu,Ludu,andHolton(2015)supportthisviewandconsidervalidlogicalreasoningasakeyelementforsoundcriticalthinking.
Otherauthorssuggestthatimprovinglogicalreasoningskillsaspartofhigherorderthinkingskillsisanimportantobjectiveofeducation(Zohar&Dori,2003).
Tosupportthedevelopmentofcriticalthinking,itseemsessentialthatteachersdevoteattentiontostudents'strategiestoreasonlogically.
Sofar,notmuchisknownaboutthereasoningprocessesofsecondaryschoolstudentsindifferentlogicalreason-ingtasks.
Therefore,thisarticleaddressesthisissuebyexploringhow16-and17-year-oldstudentsreasonwithinformalreasoningandeverydayreasoningtasks.
Theinfor-mationprovidedbythisstudyseemsimportanttoincreaseteachers'awarenessofreasoningstrategiesusedbystudentsandreasoningdifficultiestheyencounter,aswellastobeabletodevelopinstructionmaterialstosupportandimprovestudents'logicalreasoningskills.
TheoreticalBackgroundHalpern(2014)describescriticalthinkingas"purposeful,reasoned,andgoal-directed"(p.
8)andcontendsthatmanydefinitionsofcriticalthinkinginliteratureusetheterm"reasoning/logic"(p.
8),sobeingabletoapplytherulesoflogiccanbeseenasarequirementforcriticalthinking.
Manystudiesreportdifficultieswithlogicalreasoningfordifferentagegroups(e.
g.
Daniel&Klaczynski,2006;Galotti,1989;O'Brien,Shapiro,&Reali,1971;Stanovich,West,&Toplak,2016).
Becauseofthosedifficulties,itisbynomeanscertainthatsecondaryschoolstudentsareabletoreasonlogicallyandthusdeveloptheircriticalthinkingabilitiesautonomously.
FormalReasoning.
Theareaoflogiccanbedividedintoformallogicandinformallogic.
AristotlealreadydifferentiatedbetweenformallogicwithsyllogismsdescribedinAnalyticaPrioraand'dialectics'inhiscombinedworkTopicaexploringargumentsandopinions(Aristotle,2015version).
Almost2000yearslater,GottlobFrege(1848–1925)studiedanddevelopedformalsystemstoanalysethoughts,reasoning,andinferences.
Also,hedevelopedtheso-called'predicatelogic',inspiredbyLeibniz(1646–1716),whichismoreadvancedthanthe'propositionallogic'(Look,2013;Zalta,2016).
Nowadays,thosetypesofsystemsareoftencalled'symboliclogic'withstrictvalidityasakeyaspect(DePater&Vergauwen,2005)inwhichformaldeductivereasoningisapplied.
Ingeneral,formalsystemscontainasetofrulesandsymbolsandthereasoningwithinthesesystemswillprovidevalidresultsaslongasonefollowsthedefinedrules(Schoenfeld,1991).
Thecorrespondingreasoningisoftencalledformalreasoningand"characterizedbyrulesoflogicandmathematics,withfixedandunchangingpremises"1674H.
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(Teig&Scherer,2016,p.
1).
Thesameuseofformalprocedurescanbefoundindefinitionsoflogicalreasoningaswell.
Forinstance"Logicalreasoninginvolvesdeterminingwhatwouldfollowfromstatedpremisesiftheyweretrue"(Franksetal.
,2013,p.
146),and"Whenwereasonlogically,wearefollowingasetofrulesthatspecifyhowwe'oughtto'deriveconclusions"(Halpern,2014,p.
176).
However,thereisnoconsensusonthetermreasoninganditisnotexclusivelyusedforformaldeductivereasoningormathematicalsituationsonly.
Althoughreasoninginmathematicsdiffersimmenselyfromeverydayreasoning(Yackel&Hanna,2003),evenreasoninginmathematicalproofisnotonlyaformalprocedure,butinvolvesdiscussion,discovery,andexploration(Lakatos,1976)andshowsusaneedformoreinformalmethodswhenapproachingformalreasoningproblems.
InformalReasoning.
Intheprevioussection,weindicatedthat,dependentonthesituation,reasoningdemandsmorethanapplyingrulesoflogic.
Forinstance,theimportanceoftransforminginformationasstatedbyGalotti(1989):"[Reasoningisa]mentalactivitythatconsistsoftransforminggiveninformation…inordertoreachconclusions"(p.
333)andtheroleofsamenessesasstatedbyGrossen(1991):"Analogicalandlogicalreasoningarestrategiesforfindingandusingsamenesses.
…logicalreasoningappliesthesederivedsamenessesinordertounderstandandcontrolourexperience"(p.
343).
Thenotionofbroadeningformalmethodswithmoreinformalmethodsisnotnew.
ToulminalreadydiscussesthelimitationsofformallogicforallsortsofargumentsinhisfamousbookTheUsesofArgument(1958).
Hedistinguishesdifferentlogicaltypestoemphasisehowlogicisusedindifferentfields,suchaslaw,science,anddaily-lifesituations.
Inhislayoutofanargument,heschematisesthegroundsforaclaimbalancedwithreasonsthatrebutaclaim.
Healsousesqualifierstoindicatetheprobabilityofaclaim.
Philosophersandeducatorswerealsodissatisfiedwiththedominanceofformallogic,thattheyconsideredasinappropriateforevaluatingreal-lifearguments,andstartedinthe1970saninformallogicmovementforanotherapproachofanalysingargumentsstatedinordinary,daily-lifelanguage(VanEemerenetal.
,2014).
OneofthemajortextbooksstillinprinttodayisLogicalSelf-Defense(Johnson&Blair,2006),whichcoversanintroductionin"logicalthinking,reasoning,orcriticalthinking…thatfocusesontheinterpretationandassessmentof'reallife'arguments"(p.
xix).
Inliterature,thisisoftenindicatedasinformaloreverydayreasoning,butthistermhasvariousmeanings,fromreasoningoriginatingfromformalsystemstoallreasoningrelatedtoeverydaylifeevents(Blair&Johnson,2000;Voss,Perkins,&Segal,1991).
Differentfromformalreasoning,thereasoningandtheconclusionsdependonthecontextandcanbequestionedontheirvalidityasalreadyshownbyToulmin.
There-fore,thetopicsareopenfordebateandinvitetoponderonjustificationsandobjections.
Theargument,astheresultofthereasoning,oftenconcernsopen-ended,ill-structuredrealworldproblemswithoutoneconclusive,correctresponse(Cerbin,1988;Kuhn,1991).
Forthis,JohnsonandBlair(2006)use"acceptablepremisesthatarerelevanttotheconclusionandsupplysufficientevidencetojustifyacceptingit"(p.
xiii).
Theuseofacceptablepremisescanarisefrompracticalreasonstoreachacertaingoalandoftenincludespresumptionsorpresuppositions.
Walton(1996)usestheterm'presumptivereasoning'forthiskindofarguments,whichheseesasdialogues.
LogicalReasoninginFormalandEverydayReasoningTasks1675Althoughpresumptivereasoningisnotalwaysconclusiveoracceptedbyeveryone,itis,inparticulariffullknowledgeisunavailableorunobtainable,accordingtoWalton,thebestsupplementtodescribeanddiscusseverydaylifereasoning,forwhichheusesargumentationschemes.
EventhoughBlair(1999)acknowledgestheimportanceofpresumptivereasoningfordescribinghumanreasoningandthestrengthofconclusionsderivedfromthepremises,hequestionsifallargumentsaredialoguesanddiscussesthecompletenessoftheschemes.
Tosumup,wedefineinformalreasoningasreasoninginordinarylanguagetoconstructanargumentwhichrequiresacriticalreviewofthegivenpremisesandtransformingofinformation,aswellasfindingadditionalorsimilarinformationprovidedbytheproblemsolverorbyexternalsources.
TowardsaDefinitionforLogicalReasoninginThisStudy.
Now,wehaveseenthatforwell-foundedreasoning,formalandinformalmethodsareuseful,weneedtoformulateadefinitionoflogicalreasoningforthisstudy,whichcapturesbothaspects.
Adefini-tionoflogicalreasoningshouldcontainboththecontextandthewayofreasoning,whichcanconsistofformalandinformalstrategies.
Inotherwords,adefinitionoflogicalreasoningshouldnotbesynonymouswithformaldeductivereasoning.
Impor-tantkeywordstakenfromtheprevioussectionsare'deriveconclusions'fromHalpernand'transforminginformation'fromGalotti.
Thatcanbedonewithrulesderivedfromformalsystems,butthatisnotanecessity,soinformalreasoningwillalsobepartofourdefinitionandthusseenasavalidreasoningprocess.
Therefore,weconcludethatlogicalreasoninginvolvesseveralstepsanddefinelogicalreasoningforthisstudyasselectingandinterpretinginformationfromagivencontext,makingconnectionsandverifyinganddrawingconclusionsbasedonprovidedandinterpretedinformationandtheassociatedrulesandprocesses.
FormalandEverydayReasoningTasks.
Untilnow,wefocusedonthewaysofreason-ingandstressedtheimportanceofthecontext.
Ifwewanttostudyhowstudentsreasoninavarietyofcontexts,wehavetodifferentiatebetweenclosedtaskswithonecorrectanswerandmoreopentasks.
Forthis,wewilluseGalotti's(1989,p.
335)division:'formalreasoningtasks'and'everydayreasoningtasks'.
Formalreasoningtasksareself-contained,inwhichallpremisesareprovided.
Forthosetasks,establishedproce-duresareoftenavailablewhichleadtooneconclusiveanswer.
Ineverydayreasoningtasks,premisesmightbeimplicitornotprovidedatall.
Forthosetasks,establishedproceduresareoftennotavailableanditdependsonthesituationwhenananswerisgoodenough.
Indaily-lifesituations,everydayreasoningproblems"are[often]notself-contained"and"thecontentoftheproblemtypicallyhaspotentialpersonalrelevance"(Galotti,1989,p.
335).
Forbothtypesoftasks,butforeverydayreasoningtasksinparticular,selectingandencodingrelevantinformationisofgreatimportance.
Wewillcallthattheinterpretationofthetask.
Formalreasoningtasksmaybeprovidedindifferentforms:withsymbolsandcompletelyinordinarylanguagewithoutsymbols.
AsshowninFig.
1,wedifferentiateformalreasoningtasksinformallystatedandinnon-formallystatedtasks.
Formallystatedtasksarestatedwithacertainsetofsymbols,forexampleataskwiththepremises'(1)AllAareB.
(2)AllBareC.
'Non-formallystatedtasksaretasksstatedinordinarylanguage,forexampleataskwiththepremises'(1)Allmandarinsare1676H.
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oranges.
(2)Allorangesarefruits.
'Foreachtask,students'reasoningstartswithaninterpretationofthegiveninformation.
Thatmightbeeitheraformalinterpretation,inotherwords,aninterpretationwithinacertainsetofsymbols(e.
g.
ABC"AllAareC"),oraninformalinterpretationinordinarylanguage.
Everydayreasoningtasksarenottranslatabletoformalreasoningtasksandoftencontainimplicitpremisesas,forinstance,ineverydaylifestories.
Likeinformalreasoningtasks,studentswillneedtointerprettheinformationineverydayreasoningtasksaswell.
Thatcanbedonecompletelyinformally,butaformalrepresentation,suchasaschematicoverview,mighthelpstudentstogetanoverviewofthegivensituation.
Inthisstudy,wefocusbothonstudents'interpretationandthereasoningstrategiesthatfollowfromthere.
Formalisations.
Frompriorresearchamonguniversitystudents(e.
g.
Lehman,Lempert,&Nisbett,1988;Stenning,1996),weconjecturethatreasoninginallkindsofsituationswillbenefitfromtheuseofformalrepresentationsorformalisations.
Wewillusethetermformalisationinitsbroadestsense,includingallsortsofsymbols,schematisations,visualisations,formalnotationsand(formal)reasoningschemes.
Stenning(1996)givessupportfortheroleof(elementary)formalnotationsandrulesbymentioningthat"learningelementarylogiccan[emphasisadded]improvereasoningskills"(p.
227)andcanhelptounderstandformalthoughtsandarguments.
Also,Lehmanetal.
(1988)foundsupportforthenotionthatreasoningingeneralcanimproveasaresultofteachingformalruleswithinaparticularfield.
Nonetheless,thisdoesnotimplythateveryformalisationishelpful:Thechosenrepresentationshouldsupportthethinkingprocessforthespecificcontext,ratherthanthatitshouldcaptureallaspects(McKendree,Small,Stenning,&Conlon,2002).
Inthisstudy,wewillinvestigatewhichformalisationsareusedbytheparticipantsandifthoseformalisationsarebeneficial.
Fig.
1TypesoftasksandinterpretationLogicalReasoninginFormalandEverydayReasoningTasks1677ResearchQuestionsSincelittleisknownaboutthereasoningprocessesof16-and17-year-oldstudentsinlogicalreasoningtasks,ouraimistoexploretheirreasoningstrategies.
Becauseofitsexploratorynature,weselected,accordingtothedivisionprovidedinFig.
1,threeelementarytypesofreasoningtasks:twoformalreasoningtasks,tobepresentedwith(formallystated)andwithout(non-formallystated)symbols,andaneverydayreason-ingtask.
Ourexploratorystudywasguidedbythefollowingresearchquestions:(1)Howdostudentsreasontowardsaconclusioninformalreasoningandeverydayreasoningtasks,whetherornotbyusingformalisationsAnd(2)whatkindofreasoningdifficultiesdotheyencounterwhenproceedingtoaconclusionMethodsForthisexploratorystudy,weselectedclosedtasks(formalreasoningtasks)concerninglinearorderingandsyllogismsandanopen-endednewspapercomprehensiontask(everydayreasoningtask).
Theformalreasoningtaskswerepresentedformallyandnon-formally,ofwhichthenon-formallystatedtaskisacounter-itemoftheformallystatedone.
Anon-formallystatedcounter-itemisatranslationofthecorrespondingformallystatedtaskinordinarylanguageandviceversa.
Bothtaskshavesimilarconclusionsasfinalanswer,sothatthereasoningprocessescanbecompared.
Figures2and3showtheseformalreasoningtasks,bothformallystatedandnon-formallystated.
Figure4showstheeverydayreasoningtaskandthistaskdoesnothaveacounter-item.
Thisnewspapertaskisanopen-endedtaskwithimplicitpremisesandhiddenassumptions.
Inthistask,studentshavetoreconstructthelineoftheargument.
Anexpertinlogicvalidatedallitemsbycheckingwordingandcomprehensibilityofthetasks.
ThisselectionoftaskscaptureseachcategoryshowninFig.
1inwhichweexpectdifferentreasoningstrategiesandcontainsfamiliarandunfamiliartaskstoourstudents.
Foreachtask,weprovideexampleinterpretationsandsolutionsbelow.
Thesesolutionsareusedasreferencesolutionstocheckthecorrectnessofstudents'answers,but,ofFig.
2Formalreasoningtasksaboutlinearordering,formallyandnon-formallystated1678H.
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course,thereasoningtowardsaconclusioncandiffer.
Intheeverydayreasoningtaskinparticular,differentformulationsarepossible.
Thelinearorderingtasks(seeFig.
2),whichareformalreasoningtasks,have'P>S'and'PeterisolderthanSally'ascorrectanswersrespectively.
Iftakenaformalinterpretation,thereasoningcanbeP>Q>R>Sfortheorderoftheletters.
Iftakenaninformalinterpretation,youcantakeexampleagesforthefourpersons.
Forexample,ifPeteris50yearsold,thenQuintcanbe20yearsold,becausePeterisolderthanQuint.
RosieisyoungerthanQuint,soRosiecanbe10yearsold.
RosieisolderthanSally,soSallycanbe5yearsold.
Inconclusion,ifPeteris50yearsoldandSally5yearsold,thenPetermustbeolderthanSally.
Thesyllogismtasks(seeFig.
3),whichareformalreasoningtaskstoo,shouldhave'doesnotfollownecessarilyfromthegivenpremises'ascorrectanswerastheonlyvalidconclusiveoption.
Fortheformallystatedversionofthesyllogismtask,possibleformalandinformalinterpretationsarevisualisedinFig.
5.
Attheleft,thegivensyllogismistranslatedintoordinarylanguagecompletelyandthusFig.
3Formalreasoningtasksaboutinvalidsyllogisms,formallyandnon-formallystatedFig.
4Everydayreasoningtask,reasoninginanewspaperarticleLogicalReasoninginFormalandEverydayReasoningTasks1679calledaninformalinterpretation.
Inthiscase,itisexample-basedwithacounter-exampleinordinarylanguage,whichis,ofcourse,asufficientexplanationwhytheconclusiondoesnotnecessarilyfollowfromthesepremises.
However,itisimportanttorecognisethatanexampledoesnotalwaysleadtoageneralconclu-sion,inparticularforvalidsyllogisms,sointhatcase,theremustbeatranslationbacktotheformalsetting.
TheformalinterpretationwithEulerdiagramsattherightofFig.
5showsthatCdoesnotnecessarilyoverlapwithA.
Inthisinterpretation,theoriginalgivensetoflettersymbolsisused.
Similardiagramscanbedrawnforthenon-formallystatedversionofthetask.
Theeverydayreasoningtask(Fig.
4)requiresstudentsto(1)identifythepremises(reasons)leadingtotheauthor'sconclusion,and(2)tohypothesisehowthesepremisesmightbeconnectedtotheconclusionbyusinggeneralknowledgeorevidencethatmightsupporttheauthor'sconclusion.
Ourexamplesolution(seeFig.
6)isscheme-basedwithphrasesinordinarylanguage.
Weanalysesuchaschemeasaformalinterpretationinwhichthethreereasons(theidentifiedpremises)arelinkeddirectlyorindirectlytotheauthor'sconclusion.
Forthethirdreason,oneneedsanadditionalreasoningstepbymentioninganotherhiddenassumptiontomaketheargumentcomplete.
Weassumethatthereissufficientgeneralknowledgeonthissubjectamongtheparticipants.
Thearrowsrepresentif-thenstatementsandarenotonlypartoftheformalscheme,butalsoformalisationsinthemselves.
Nevertheless,theif-thenstatementsintheschemecanbeexplainedinfullsentencestoo.
Forthefirsttworeasons,thatwilllooklike'Ifpeoplesmokeorinhaleparticulatematter,thenitwillaffecttheirhealthandthusshortentheirlife.
'Suchconsiderationsbasedoncommonknowledgestillshowtheconnection,butitisnotyetformalised,neitherwithascheme,norwithanysymbolsandthusconsideredasacompletelyinformalinterpretation(seeFig.
1).
Assoonasoneintroduceslogicalsymbols,wewillcallthosesymbolsformalisations.
Incombi-nationwiththeif-thenrule,thesentencecanberepresentedas'(smoking∨inhalingparticulatematter)unhealthyshorterlife'.
Fig.
5Formalandinformalinterpretationsoftheformallystatedsyllogismtask1680H.
Bronkhorstetal.
ParticipantsOurparticipantsareDutchsecondaryschoolstudentsintheirpenultimateyearofpre-universitysecondaryeducation(11thgraders)andvolunteeredtoparticipateinthink-aloudsessions.
Thefirstauthorofthisarticlewastheirteacherandtheyallsignedaninformedconsent.
Thesestudentsdidnottakeadvancedmathematicsorscience,butfollowedamathematicscourseinwhichlogicalreasoninghasrecentlybecomeacompulsorydomain(CollegevoorToetsenenExamens,2016).
Thisstudywascon-ductedbeforetheparticipantsreceivedteachinginlogicalreasoning.
Inthisarticle,workisdiscussedfromtwomale(Edgar,James)andtwofemalestudents(Anne,Susan).
ProcedureWeconductedtask-basedinterviewsinwhichstudentssolvedlogicalreasoningtasksaloud(Goldin,2000;VanSomeren,Barnard,&Sandberg,1994).
TheinterviewswereconductedinDutchandrecordedwithasmartpensothatverbalandwritteninforma-tioncouldbeconnected.
Thestudentswereaskedtosayaloudeverythingtheywerethinkingof.
Theinterviewer,whoisthefirstauthorofthisarticle,refrainedfromcommentingasmuchaspossible,sothat'freeproblem-solving'wasakeyaspectofthesessions.
Ifastudentdidnotunderstandthetaskorthoughtitwasdone,theinterviewerwouldaskadditional(clarification)questions,butdidneverprovidefeedbackonthegivenanswers.
AnalysisThetranscriptsoftheinterviewswereanalysedinDutchandselectedpartsweretranslatedtoEnglishforthisarticle.
Students'tasksolvingwasanalysedqualitativelyinaninterpretivewayanddata-driven(Cohen,Manion,&Morrison,2007).
Togetaclearpictureofthereasoningprocess,thedatasources,interviewtranscriptsandstudents'writtennotes,wereanalysedaccordingtoourdefinitionoflogicalreasoning.
Ouranalysisincludedthefollowingsteps:(1)students'understandingofthetask,(2)students'interpretationofthetask,(3)students'reasoningprocessandstrategiesused,(4)students'useofformalisationsand(5)thecorrectnessofstudents'finalanswers.
Ifstudentsswitchbetweeninterpretations,wewillcallthepredominantinterpretation,theirmaininterpretation.
Fig.
6FormalschemefortheeverydayreasoningtaskLogicalReasoninginFormalandEverydayReasoningTasks1681Students'reasoningincounter-itemsisintendedasanexplorationofpossiblevariationinreasoning.
Becausestudentsworkedononlyoneofeachtwocounter-items,wecannotanalysethedifferencesbetweenindividualstudents'strategiesonalternativeversionsofsimilarclosedtasks.
Tojudgethecorrectnessoftheirfinalanswers,students'writtennotes,aswellastheinterviewtranscripts,areusedandcompared.
Possibledifferencesaremarkedandcombinedwiththeirinterpretationsandreasoning.
Wehavetonotethattheverbalexplanationsinitselfcanbeseenasinformal,becauseifstudentsareaskedtodotasksaloud,theyuseordinarylanguage,butifexplainedwitha(given)setofsymbols,theinterpretationofthetaskcanstillbeformal.
Furthermore,theverbalexplanationsarelinkedtowrittennotes,inwhichpossibleuseofformalisationsisclearlyvisible.
ResultsTable1providesanoverviewoftheresults.
Thereafter,foreachtaskstudents'reasoningwillbeillustratedindetail.
ReasoningwithLinearOrderingFormalreasoningtaskswithlinearordering(seeFig.
2)arefamiliartothestudentsbecausethesetypesoftasksarecommoninprimaryandsecondaryeducation.
WesummarisetheTable1Overviewofstudents'interpretations,strategies,formalisationsandcorrectnessanswersTaskTypeoftaskStudentMaininterpretationStrategiesFormalisationsCorrectanswerLinearorderingFormal:FormallystatedEdgarFormalRule-basedGivensymbolsYesAnneInformaltransformationRule-based–YesFormal:Non-formallystatedSusanFormalRule-basedLetterabbrSymbolsNumberlineYesJamesInformalRule-based–YesInvalidsyllogismFormal:FormallystatedSusanFormal–SymbolsArrowsNoJamesInformalAnalogy-/Example-based–NoFormal:Non-formallystatedEdgarInformalRule-based–YesAnneInformalRule-based–YesAnalysisnewspaperarticleEverydayreasoningtaskSusanInformalScheme-basedBulletsArrowsYes*AnneInformalInformalBulletsYes**Writtenanswerincomplete(Susan)andonlyafterclarificationquestion(Anne)1682H.
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findingsfirst:Allfourstudentsusedrule-basedstrategies,buttheirinitialinterpretationdiffered.
Allanswerswerecorrectandwell-reasoned.
Onlyonestudentcameupwithanadditionalformalisationotherthanthegivensymbols.
Sheusedaverysuitabletool,anumberlinerepresentationwithformalletterssymbols,togetaclearoverviewoftheorder.
Wewillpresentadetaileddescriptionofthefourstudents.
FormallyStated,Edgar.
Edgarinterpretsthetaskinaformalwaybycopyingtheformalnotation,seefirstthreelinesinFig.
7.
Afterwritingthatdown,hisfirststatementsareswitchingtoexample-basedreasoning(informalinterpretation)thatinvolvesfillinginsomenumbers(line[1]intranscript).
Afterthat,hequicklyweighshistwointerpre-tations(lines[2]and[3])andswitchesbacktotheformalsituation,bycomparingthegivenlettersP,RandSwiththesymbolfor'greaterthan'(line[4]andFig.
7).
Althoughtheverbalexplanationsareinwords,inherenttothinkingaloud,hesolvesthetaskbyfollowingmathematicalrulesbystayingintheformalsystemwiththecorrespondingformalsymbols.
Thiswayofreasoningprovidesthecorrectanswerquicklyandusingthegivensymbolsonlygivesaclearstructure:P>R,R>S,P>S.
Edgar:[1]well,yes,youcouldjustfillinnumbersofcourseasanexample,[2]wellohno,let'swait[3]wearenotgoingtodothatatfirst[4]uhm,PisgreaterthanQ,soPisalsogreaterthanR,…FormallyStated,Anne.
Afterreadingthetask,Annestartsimmediatelywithatransla-tionoftheformalsymbolsintoexpressionsinordinarylanguagebywritingdown'greaterthan'and'lessthan'infull,thusgivinganinformaltransformationofmostoftheformallystatedtask(seeFig.
8).
AlthoughshestillreasonswiththegivenformalFig.
7Formallystatedlinearorderingtaskattheleft,Edgar'swrittennotesattherightPRPRSogrelesgregreothatersthatereaterherefrthahanqrtharthforeanqqanRanSePiqRSisallsoggreaaterrthaanSSFig.
8Anne'swrittennotesattheleft,EnglishtranslationattherightLogicalReasoninginFormalandEverydayReasoningTasks1683letters,sheswitchestoordinarylanguageforapplyingthemathematicalrules.
Sheprovidesthecorrectanswer.
Non-formallyStated,Susan.
Susantranslatesthenon-formallystatedversionofthisformalreasoningtaskimmediatelyintoaformalsituationwithletterabbreviationsforthenamesandthesymbols>andBesidestheseformalsymbols,sheputsthelettersinsequentialorderhorizontally,whichcanbeseenasa'numberline'representationwithformallettersymbols,startingwithP-Q-RreasoningthatQmustbeinthemiddle,seeFig.
9.
Wecallthatanotherformalisation.
AfteraddingSaswell,shecomestotherightconclusionthatPetermustbeolderthanSally,whichisatranslationfromherformalsystemtotheconclusionaskedforinordinarylanguage.
Non-formallyStated,James.
Jamesreasonsinwordswithinthenon-formallystatedversionofthistaskleadingtoacorrectconclusion.
Wecallhisinterpretationinformalwithacorrectapplicationofmathematicalrules.
Aftertheconfirmationthathehastowritehisreasoningdown,hiswrittenexplanationiscompletelyinordinarylanguage,usingthegivennamesandthephrases'olderthan'and'youngerthan'(seeFig.
10).
So,James'sinterpretationiscompletelyinformalwithoutswitching.
ReasoningwithanInvalidSyllogismFormalreasoningtaskswithsyllogisms(seeFig.
3)areunfamiliartaskstothesestudentsbecausetheyarenotusedtoreasoningwithinsyllogisms.
Wesummarisethefindingsfirst:Threeofthefourstudentsusedaninformalinterpretation,butonlytwostudentsprovidedacorrectanswer.
Theformallystatedversioncauseddifficultiesduetonotunderstandingthetaskorduetoincompletetranslationstoaninformalexample.
Also,themisinterpretationof'are'andtheconfusionbetween'all'and'some'arenoteworthy.
Wealsofoundthatarecognisablenon-formallystatedcontextcansupportthereasoning,despitesomehindranceofreal-lifeexperiencesconcerningthecontextaswell.
Wepresentadetaileddescriptionofthefourstudents.
FormallyStated,Susan.
Susanshowsthatsheunderstandsthatshehastoacceptthetwopremisesinthisformalreasoningtask,regardlessoftheirtruthsbywriting"true"behindit,seeFig.
11.
Hernextstepisformalisingthegivenstatementsevenfurtherbyintroducingtheequalitysign,seefirstlinesinherwrittennotesinFig.
12,sosheinterpretsthetaskcompletelyformally.
Susantriestoreasonwiththegivenlettersfourtimes(seefoursectionstranscript)beforeshegivesup.
Again,herverbalexplanationsareinordinarylanguage,ofcourse,inherenttothinkingaloud,butsheusesthegivenlettersandstaysintheformalsystem,sowecallthataFig.
9Susan'swrittennotes1684H.
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formalinterpretation.
Inherfirsttry(lines[1]–[7]intranscript),shestatesthatAandBareequal(line[5]),butshecannotconnectthiswithC.
Inhersecondtry(lines[8]–[14]),shestartswithstatingthatAandBareequal,butcannotconnectCwiththatalthoughsayingthatsomeBarenotC(line[10]).
Inherthirdtry(lines[15]–[17]),shesays,oncemore,thatAandBareequal,butshecannotconnectthatwithC,becauseshedoesnotknowwhichB'sareC.
ThefourthtimeshewritesdownthelasttwolinesshowninFig.
12,connectingsomewithasymbolforapproximately,butthatdoesnothelpeither(lines[18]–[25]).
Itisimportanttonoticethatsheusestheequalitysigneachtimeasequaltowhichconflictswiththeoriginalpremisecontaininganinclusion.
Afterunderliningherconclusion'A≈C'inthefourthtry,shegivesupandsighs:"Ijustdonotunderstandthelogicofallthis"(line[25]).
Susanonlyreasonedwiththegivenlettersandformalsymbolsanddidnotswitchtoaninformalsituation.
Susan:[1]allAareB,…,soisequal[2]butsomeofthoseareC[3]sosomearenot,someAarenoteither,someAare[4]…mmm…[5]allAareB,soAandBareequal[6]someBareC,sosomeBareonlyA[7]andsomeBareC…mmm…[startreasoningfromthebeginningagain][8]allAareB,AandBareequal[9]someofthoseareCandsomearenot[10]someBarenotC[11]someA,thatisalsoB[12]someB…someAareC[13]…butallAareB,andsomeBareC,someAareC[rereadinggivensyllogism][14]no,Idon'tthinkso[startreasoningfromthebeginningagain][15]Ithinkthat,…uhm…,ifallAareB,AandBareequal[16]butsomeBareC,sosomeofthoseB's,thathastobethecase,donotnecessarilyhavetobeA,becauseyoudonotknowwhichB'sareC,becausethoseareequaltoC,andAandBareequal,someAareCRobuansoosieutyondPoPeeisounPeteeteroldngerralrisaerthrthareadalsohananQdywoldnSalQuinwasdertllyntoldthandertnSathanallynQuuintFig.
10James'swrittennotesattheleft,EnglishtranslationattherightPrPrCoremremoncmisemiseclusi1:A2:Sion:AllASomSoAarmeBomereBBareAaB.
eC.
areC.
C.
truueFig.
11Susanacceptsthegivenpremises,EnglishtranslationattherightLogicalReasoninginFormalandEverydayReasoningTasks1685[17]ow,Ireallythinkthisisdifficult…[startreasoningfromthebeginningoncemore][18]okay,allAareB[writesdownA=B][19]someBareC,soapproximately[writesdownB≈C][20]andsomeAareC,butAandBareequal[21]someofthoseBareC[writesdownbehindA=B:B≈C][22]andsomeAareC[writesdownA≈C][23]so,myconclusion,…mmm…[underlinesA≈C][24]Ireallydon'tknow[25]IjustdonotunderstandthelogicofallthisFormallyStated,James.
Jamesrecognisesthathedoesnotknowhowtosolvethistaskinaformalwaybyexpressing"Idon'tknow",soheswitchestoaninformalinterpretationoftheformallystatedtask:startingwithsearchingforanexampleinordinarylanguage.
Thiscanbeseenasanalogy-andexample-basedreasoning.
HisexplanationiscloselyrelatedtoourexampleinFig.
5,butJamesonlylooksforonevalidexampleinsteadofacounterexample.
Hechoosesanexampleinwhich'some'representsallapes(line[3]intranscript),becausethesetofapesnotbeingmammalsisempty.
Weassumethathedidnotrecognisethatbecauseofhisincorrectconclusion.
Hetriestousealogicalstructure'if-then'(lines[2]–[4])aswell,butthatdoesnotsolvetheproblem.
Afterthevalidconclusionofhisexampleinordinarylanguage,hetriestoexplainthevalidityofhisconclusioninamoreformalwaywiththegivenletters(lines[5]–[7])andwritesthatdownaswell(seeFig.
13).
Forthis,JamesalsostatesthatAandBareequal(line[5])inthesamewayasSusandid,andisnotabletoprovideamorepreciseexplanationafterclarificationquestionsbytheinterviewer.
James:[1]okay,well,IamgoingtohavealookwithasimilarexampleIthink[2]if,uhm,allhumansareapes[3]someapesaremammals[4]thensomehumansare…alsomammals[5]so,Ithinkitiscorrect,becauseAandBareequal,Fig.
12FormalisationsusedbySusan,EnglishtranslationattherightASoSollAoifomeA'sasomeA'arenmeB'sarnecB'sreCessaareC'starilyC'stoo.
yBs.
Fig.
13James'swrittennotesattheleft,Englishtranslationattheright1686H.
Bronkhorstetal.
[6]becausethatisnecessarilytrue,[7]soifthat'sthecaseforsomeB,itisalsothecaseforANon-formallyStated,Edgar.
Edgar'sinterpretationofthenon-formallystatedversionofthisformalreasoningtaskisinformal.
Hedrawsthecorrectconclusionquiteeasily(line[2]intranscript).
Healsoexplains,althoughthisisnotnecessarilytrue,thepossibilitythatsomeflowersmightrefertorosesaswellastootherflowers(line[6]),whichshowsanotionoftherulesoflogic.
Inhiswrittennotes(seeFig.
14)healsoshowsthatthewordflowercouldcontainmorethanonetypeofflowers.
Thisreflectionisquitestrongandshowsinsightinthegeneralityofasyllogism.
Edgar:[1]uhm…well…let'ssee…[2]yes,youwouldsaythatthisdoesnotfollowlogically,becausesomeflowersdoesnotnecessarilyrefertoallroses[3]let'ssee[4]some,yes,[5]uhm…[6]itdoesnothavetomeanthatrosesfadequicklysincesomeflowersmightalsobedaisiesor,well,something,orotherflowersconsequentlyNon-formallyStated,Anne.
Annealsodrawstherightconclusioninthenon-formallystatedversionofthistask.
Sheusesaninformalinterpretationandcomesupwithacorrectanswerquickly(line[1]intranscript)andprovidesamorecompleteexplanationinhernextsentence(line[2]),whichissimilartoherwrittenanswer(seeFig.
15).
However,Anneisnotcompletelysureaboutheranswer.
Askedforanexplanation,shesaysthatheruncertaintycomesfromherknowledgeaboutfadingflowers(line[8]and[9]),althoughsherecognisesthatonecannotconcludethatfromthesepremises,whichshowsthatsheunderstandstherulesoflogic.
Anne:[1]Youdonotknowifit'stherosesthatfade,soyoualsodon'tknowifsomerosesfadequickly.
[2]Allrosescanstillbeflowers,andsomeflowerscanstillfadequickly,butthatdoesnothavetomeanthatroses[sighs]fadequickly.
[3]Yes,Ithinkso.
[4]Iamlesscertainaboutthisone.
…[5]becauserosescanstillbeflowers,but,owwait,and[6]…thatdoesnothavetomeanthat,perse,therosesfadequickly,…Intvwr:[7]AndwhyareyoulesssurethanintheprevioustaskSodicoomeirectouldeflotlytdpoowetoroossibersdosesblybdons.
BbernotrButsroserefersomes.
rmeflloweersFig.
14Edgar'swrittennotesattheleft,EnglishtranslationattherightLogicalReasoninginFormalandEverydayReasoningTasks1687Anne:[8]uhm…yes,because,someflowersfadequickly,yes,Idon'tknow,Iknow,Ithinkit'sdifficulttoexplain,butIamjustmoreindoubthere,because…[9]IwasthinkingbecauseIknow,ofcourse,thattherearemanyotherspeciesofflowersthanonlyroses,onlyfromthesepremisesyoucannotseethatofcourseReasoninginaNewspaperArticleAneverydayreasoningtaskabouttheanalysisofanewspaperarticle(seeFig.
4)isconsideredunfamiliartothesestudents.
Wesummarisethefindingsfirst:Bothstudentsusedinformalinterpretationsandonlysomebasicformalisationstosumupreasonsormakeconnectionseventhoughoneofthestudents(Susan)providedtosomeextentaschematicoverview.
Althoughnotessential,theydidnotbuilda(strong)formalschemeas,forexample,showninFig.
6.
Wepresentadetaileddescriptionofthetwostudents.
Susan.
Susanstartsthistaskwithidentifyingthethreepremises(step1)inastructuredwaybywritingdownthethreereasonsmentionedinthearticlebehindbullets(seefirstthreelinesFig.
16).
Thereafter,shetakeshertimetoreconsiderthesereasons,thewordingofthetaskandthephrase'hiddenassumption'.
Shewritesdown"thehiddenassumptionisthatpeoplefromRotterdamlivelesshealthy",whichhypothesiseshowthepremisesarelinkedtotheconclusion(step2).
Sheexplainsthat"ithastodowithpeople'shealth"becauseofthefirsttworeasons,smokingandworseenvironment,butSusanstruggleswithanexplanationforthethirdreason:lowereducationandincomelevel(seeline[1]intranscript).
Thisreasondemandsmoreevidence.
Susanimpliesthatpoorerfamiliesarethemissingconnectionforthelowerincome(line[4]).
Forthat,sheusesanotherformalisation,whichstructuresherwrittennotes:anarrowtomaketheconnection.
Verbally,sheprovidesafurtherexplanationfortheassumption'poorerfamilies'(line[5]),butshedidnotwritethatdown.
Asotollroomeomeosesefloeanscanowerthatnstrscatrostillbanfsesbeflfadenecflowequcesswersuicklarilyandly,tyfadthatadetdoquicesncklynothy.
haveFig.
15Anne'swrittennotesattheleft,EnglishtranslationattherightRehidlivThpopeeasonddenvelesheshoorereoplehiddns:nassusshehorterrheaedena-n-h-lumpealthrlifelthinassumnumhighloweptionyeiscnRomptiomberohercoeredisthcauseotterdonofsmonceducathatpsmoworlowpooedbydammokeentrattionaeoplokinrseewereorerfythecomerstionalandefrongenvireducafamiempareofpadincomRronmationiliesedtoarticcomeRottementnalanotheculateelevrdamndinerDuemavelmncomutchattermelevelFig.
16Susan'swrittennotesattheleft,Englishtranslationattheright1688H.
Bronkhorstetal.
Overall,inherverbalexplanation,shehasconnectedallthementionedreasonswithahiddenassumptionleadingtohermainassumption'poorerhealth',whichsheunderlinesaswell.
Inherwrittennotes,sheusesformalisationsatthreemoments:atthebeginning(bullets)forthefirststepinvolvingidentifyingthepremises,andarrowsattwotimesforconnections,eitherwithhypothesisedevidencebasedonherownknowledge(step2),ortoemphasisethemainhiddenassumption.
Hernotesprovidemoreorlessaschematicoverview,butSusandidnotcompileacompleteformalscheme.
Susan:[1].
.
.
mmm,so.
.
.
thelowereducationandincomelevelwhatdoesthathavetodowith.
.
.
lowerlevelofeducation,.
.
.
mmm.
.
.
,yes,theamountofsmokershastodowithhealthandthehighconcentrationofparticulatematterintheair,soitmeansthatthehealthofpeoplefromRotterdamisworsethanthehealthofotherpeopleintheNetherlands[2]uh,tolink,explainhowthereasonsmentionedarelinkedtotheshorterlife,bydescribingthehiddenassumption,uhm.
.
.
[3]Theshorterlifeiscausedby.
.
.
the.
.
.
poorerhealthinRotterdamcomparedtootherDutchpeople.
[4]Then,thehiddenassumptionisthatpoorerhealthand.
.
.
maybe,uh.
.
.
lowereducationandincomelevel,soperhapspoorerfamilies[drawsanarrowtoconnectthiswithlowereducationandincomelevel][5].
.
.
andtheymaynotbuyveryexpensiveandorganicfoodandeverything,sotheywouldlivelesshealthyorsomething,Ithinkthehiddenassumptionisthattheyeatlesshealthy,orlivelesshealthylivesespecially,yesthat'swhatIthink[6]ThisiswhatIthink,thepoorerhealth[underlinespoorerhealth],that'sthehiddenassumption.
[addsarrow]Anne.
Anneunderlinesthethreemainreasonsinthetext:smokers,particulatematter,lowereducationandincomelevel,whichshowsthatsheidentifiedthepremises(step1).
Afterthat,sheliststhethreereasonsbehindbullets(seeFig.
17).
Thatistheonlyformalisationsheuses.
Therestofthereasoning,verbalandwritten,isdoneinordinarylanguage.
Forthesecondandthirdreason,sheprovidesahiddenassumption:"partic-ulatematterisbadforsomeoneandthusshortensone'slife,andthelowereducationlevelandthelowerincomelevelleadstopoorerlivingconditionsandthusshortensone'slife".
Sheusesherownknowledgetostatethatparticulatematterisbadforsomeone'shealthandtohypothesisethatalowerincomelevelleadstopoorerlivingTtlTheTheandThethuslivin'shorassuassuthusassuthengcorteneumptumptshorumptloweonditedliftionttiontrtensionterincionsfe';sthatthatpsyouthattcomandshortbadsmokpartiurlifthelelevthusterthdforkingiculafeowervellesshohaniyougsatemredueadsorteninthuandshortmatterucatitoponsyoeresdthustensyrisbionleooreurlistofsyourbadfevelerfe.
theNrlifeforyandNethouherlanndsFig.
17Anne'swrittennotesattheleft,EnglishtranslationattherightLogicalReasoninginFormalandEverydayReasoningTasks1689conditions(step2).
However,sheforgetstoprovideaconnectionforthefirstreason,sotheintervieweraskedforfurtherexplanationsbeforesheadded"badforyouandthus"(seetoplineFig.
17)fortheconnectionbetweensmokingandshorteningone'slife.
Overal,Anneidentifiedthepremisesquitequicklyandprovidedsupportforthereasonseasily.
Probably,sheassumedthattheconnectionbetweensmokingandashorterlifewasgenerallyknown,sothatsheonlyprovidedadditionalevidenceafteraclarificationquestionbytheinterviewer.
ConclusionsandDiscussionThepurposeofthisstudywastogaininsightsintothereasoningprocessesof16-and17-year-oldpre-universitysecondaryschoolstudentsonlogicalreasoningtasks,aimedatfosteringtheircriticalthinkingskillsasanimportantobjectiveinthetwenty-firstcenturyskillsframework(P21,2015).
Inthisexploratorystudy,weinvestigated(1)thewayofreasoningstudentsusedinformalreasoningandeverydayreasoningtasksandtheiruseofformalisations,and(2)thedifficultiestheyencounterintheirreasoning.
Inthelinearorderingtasksandinthesyllogismtasks,studentsusedrule-,analogy-andexample-basedreasoningstrategies.
Inthenewspaperarticletask,studentsrea-sonedpartlyscheme-based,butmainlyinordinarylanguageonly.
Exceptfortheformallystatedsyllogismtask,studentsusedappropriatestrategiestofindcorrectanswers.
Althoughthelinearorderingtasksarefamiliartothestudents,bothformats(formallyandnon-formallystated)ledtoformalandinformalinterpretations.
Studentsdonotalwaysfeelcertainabouttheirmethodandanswer,inparticularinthesyllogismtasksandintheeverydayreasoningtask.
Theincompletewrittenanswersintheeverydayreasoningtaskshowthatourstudentsprobablyhavedoubtsiftheiranswersaregoodenough,becausetheyhavethefeelingthatmoreanswersarepossible.
Eventhoughthenewspapertaskistakenfromaneverydaylifecontextand—weexpect—recognisableandmeaningful,studentsstillfulfilataskforwhichtheyexpectthatthereshouldbeonecorrectanswer,asiscommoninmathematicstasks(e.
g.
Jder,Sidenvall,&Sumpter,2017).
ThedoubtstudentsexpressisinlinewithGalotti's(1989)descriptionforeverydayreasoningtasks,becauseshestatesthat"itisoftenunclearwhetherthecurrent'best'solutionisgoodenough"(p.
335)incontrasttoformalreasoningtaskswhere"itistypicallyunambiguouswhentheproblemissolved"(p.
335).
Inourformallystatedsyllogismtask,thestudentsmisinterpretedthephrases'all…are…'and'some…are…'.
Consequently,theydidnotseethattheirrepresentations,suchastheuseoftheequalitysignasaformalsymbol,werenotsuitable.
Susanwascompletelystuckintheformallystatedversionandcouldnotfindawayout.
Themisuseoftheequalitysign(=)for'all…are…'isacommonmistake(e.
g.
Galotti,1989,p.
336).
StenningandVanLambalgen(2008)alsodescribedifficultieswithunderstandingandinterpretingsyllogisms.
Anoverviewofourfindingsisvisualisedinascheme(Fig.
18)asanextensionofFig.
1.
Weshowedthatstudents'initialinterpretations,theirfirstthoughts,donotalwaysmatchwiththeirlaterchoices,sostudentsseemtoswitchbetweenformalandinformalinterpretations.
Thisisvisualisedbythearrowinthescheme.
Thestrategies1690H.
Bronkhorstetal.
includedinthisschemearederivedfromourexploratorystudyamong16-and17-year-oldstudentsandmightnotprovideacompleteoverview.
Consequently,theoverviewcanbesupplementedwithargumentationschemesbasedonpresumptivereasoning(Walton,1996;Walton,Reed,&Macagno,2008)infurtherresearch.
Oureverydayreasoningtaskgivesonlyalimitedviewofstudents'possiblereasoningstrategies,becausestudentswereonlyaskedtoidentifythepremisesandtousetheirownknowledgetofindconnectionswiththeconclusion.
Theywerenotaskedtofindrebuttalsorfurtherbackingoftheclaims.
Theconnectionsprovidedbythestudentsshouldbesufficientforjustifiablereasoning.
ThiscorrespondentstotheearliermentioneddescriptionfromJohnsonandBlair(2006)aboutacceptablepremises.
EachofthestrategiesshowninFig.
18canbesupportedbytheuseofformalisations.
Forexample,inthenon-formallystatedlinearorderingtask,Susanusedletterabbreviations,mathematicalsymbolsandanumberlinerepresentation.
Forcasesinwhichstudentsreasoninordinarylanguagewithoutclearlyshowingcausality,comparisonorexamples,weaddedthecategory'informalreasoning'.
ThiscategoryisbasedonourdefinitionofinformalreasoninginthecorrespondingsectionintheTheoreticalBackground.
Webelieveitisimportanttopresent'informalreasoning'asseparatecategoryinthescheme,becausestudentsstillmanagedtoconstructanargumentinordinarylanguage,butwithoutclearlyshowingavisiblereasoningstrategy,suchasrule-based,example-based,scheme-based,etcetera.
.
Therefore,weusedadottedlineinFig.
18.
Consequently,inthatcase,formalisationscanonlybeusedtoacertainextentas,forexample,shownbyAnneinheranalysisofthenewspaperarticlewheresheseparatedherthreeinformalargumentsbybullets.
Inthisarticle,wehypothesisedthatsuitableformalisationscansupportthereasoningprocessandsummarisedthosetoolsattheright-handsideofFig.
18.
Webelievethatourhypothesisisstrengthenedbythefindingsinthisexploratorystudy.
Symbols(like'greaterthan'and'lessthan',ortheequalitysign)andletterabbreviationsaresuitabletoolstoshortennotations,whileothertools(likeanumberlinerepresentation)arestrongtoolstovisualiseinformation.
Althoughnotusedbyourstudents,VennandEulerdiagramsarealsostrongtoolstovisualisedata.
However,itisourconvictionthatFormalreasoningtasktaskwithfixedandunchangingpremisesFormallystated:statedwithacertainsetofsymbolsFormalinterpretation:interpretationwithinacertainsetofsymbolsRule-based:applyingrulesoflogicand/ormathematicsbasedonprovidedpremisesExample-based:providingvalid-and/orcounter-examplesAnalogy-based:lookingforsimilarities/samenessesFormalisations:-Letterabbreviations-Bullets-(Logical)symbols-Venn-/Euler-diagrams-Numberline-Arrowsforinferences-EtceteraScheme-based:using(formal)scheme(s)Informalreasoning:plainargument(s)inordinarylanguageInformalinterpretation:interpretationinordinarylanguageNon-formallystated:statedinordinarylanguageEverydayreasoningtaskopen-endedtask,oftenwithimplicitpremisesTypeoftask:Interpretation:Strategies:Fig.
18Typesoftaskscombinedwithstudents'interpretationsandreasoningstrategiesLogicalReasoninginFormalandEverydayReasoningTasks1691theuseofformalisations,includingvisualisationssuchasVennandEulerdiagrams,isteachableandcanbelinkedtothestrategiesusedbythestudents,alsoineverydayreasoningtasks.
Alimitationofthestudyisrelatedtoourchoiceoftasks.
Inourselectionoftasks,weusedformallystatedtasksandnon-formallystatedtasksascounter-itemsforsimilarreasoningproblems.
Inourdesign,differentstudentsworkedononeofthecounter-itemsandtherefore,wecouldnotcomparetheperformanceofanindividualstudentonbothtasks.
Thenon-formallystatedtasksweremoreeasilytointerpretbythestudentsandledtootherstrategies,becausetheirpriorknowledgewashelpful.
Hintikka(2001)explainsthat"inreal-lifereasoning,evenwhenitispurelydeductive,familiaritywiththesubjectmattercanbestrategicallyhelpful"(p.
46).
Ontheotherhand,sometimesourstudentsmaydoubttheiranswers,becausepremisesinthetask(e.
g.
Anneinthenon-formallystatedsyllogismtask)mightconflictwiththeirpriorknowledge.
Ingeneral,thismeansthatourcounter-items(formallyversusnon-formallystatedtasks)cannotbeconsideredasequivalent.
Despitethefactthatourstudyhasalimitationinthenumberofparticipants(smallandselectivesample)andalimitednumberoftasks,theinformationinFig.
18showsavarietyofreasoningstrategies,whichisimportantforteacherstounderstandthediversityofstudents'reasoningandpossibledifficultiesintheinterpretationsoftasks,inparticularfortasksthatarenotfamiliartostudentsorleadtoincorrectanswers.
FutureResearchandRecommendationsThisstudynotonlyshowsthecomplexmatterofreasoningandeverydaylifereasoninginparticular,italsoconfirmsthatmoreresearchisneededasalreadymentionedbyGalotti(1989,2017).
Ourexploratorystudyisafirststeptogetinsightsinthereasoningprocessof16-and17-year-oldpre-universitystudentsandshowsagapbetweentheirverbalandwrittenexplanations.
Wewillcontinueourresearchforanin-depthunderstanding.
Unfamiliartasks,suchasallsortsofnon-formallystatedsyllo-gisms(formalreasoningtasks)andeverydayreasoningtasksseemtobeusefulcontextstoinvestigatehowstudentssolvereasoningtasksandwhichformalisations,includingvisualisations,theyuse.
Ourdefinitionoflogicalreasoning,mentionedintheTheoret-icalBackground,fitsthisfutureresearch.
Ourresultsshowthatstudentsdonotstructureeverydaylifecontextsautomatically,soitisplausiblethatsimilardifficultiesoccurinauthenticeverydaylifereasoningtoo.
Infutureresearch,weintendtoshowthatstudentsmaybesupportedbylearningmorestructuredreasoningstrategiesandtheuseofformalisationsandvisualisations.
Oneofthekeyaspectsforlessonsinlogicalreasoningmustbeclassroomdiscoursewhensolvingreasoningtasks.
Lakatos(1976)alreadystressedtheimportanceofdialogueintheconstructionofmathematicalandlogicalreasoning.
Ourresearchmightincreaseteachers'awarenessofthatimportanceand,morepractically,forwhichFig.
18servesasaguidelinefordiscussion.
Differentinterpretationsandpossiblestrategiesusedbystudentsaremadeexplicitandcanbeusedasinputforclassroomdiscussions.
Wesuggestthatformalisationsandvisualisationsarepartofthosediscussionsandmightestablishadeeperunderstanding.
Aboveall,logicalreasoningtaskswhereseveralwaysofreasoningarepossible,arehighlyconnectedtothetwenty-firstcenturyskills(P21,2015),andthuswiththedevelopmentofcriticalthinkingskills.
1692H.
Bronkhorstetal.
AcknowledgementsThisworkispartoftheresearchprogrammeDoctoralGrantforTeacherswithprojectnumber023.
007.
043,whichis(partly)financedbytheNetherlandsOrganisationforScientificResearch(NWO).
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