education sciences Article Multimodal Languaging as a Pedagogical Model—A Case Study of the Concept of Division in School Mathematics JormaJoutsenlahti*andPirjoKulju SchoolofEducation,UniversityofTampere,Tampere33100,Finland;pirjo.kulju@uta.fi * Correspondence:jorma.joutsenlahti@uta.fi;Tel.:+358-505-439-197 AcademicEditor:EilaJeronen Received:31October2016;Accepted:28December2016;Published:4January2017 Abstract: Thepurposeofthisstudyistopresentamultimodallanguagingmodelformathematics education. The model consists of mathematical symbolic language, a pictorial language, and a naturallanguage. Byapplyingthismodel,theobjectivewastostudyhow4thgradepupils(N=21) understand the concept of division. The data was collected over six hours of teaching sessions, during which the pupils expressed their mathematical thinking mainly by writing and drawing. Theirproductions,aswellasquestionnaireaftertheprocess,wereanalyzedqualitatively. Theresults showthat,inexpressingthemathematicalprobleminverbalform,mostofthestudentssawitas a division into parts. It was evident from the pupils’ texts and drawings that the mathematical expressionofsubtractioncouldbeinterpretedinthreedifferentways. Itwasfoundthatthepupils enjoyedusingwritinginthesolutionofwordproblems,anditissuggestedthattheuseofdifferent modes in expressing mathematical thinking may both strengthen the learning of mathematical conceptsandsupporttheevaluationoflearning. Keywords: multiliteracy;languaging;division;mathematics;education 1. Introduction Traditionally,therehasbeenastrongemphasisontheuseofsymbolicmathematicallanguage in representing mathematics. This has, however, been found to be a limiting factor in expressing mathematicalthinkinginlearningprocesses[1–3]. Theaimofthisarticleistopresentamultimodal languaging model, in which the ways to express mathematical thinking are expanded beyond mathematicsymboliclanguage. Asecondobjectiveistoobservehow4thgradepupilsunderstandthe conceptofdivisionbasedonthismodel. Differentwaysofexpressingthinkingandmakingmeaningformtheunderlyingtheoreticalbasis forthisstudy. Theoretically,themultimodallanguagingmodelisrelatedtomultiliteracy,aconcept referringtovariousmodesinthecurrentcommunicationsenvironment[4].Thepresentmodelincludes threetypesofsemioticsystemsofmeaning-making: asymbolicmathematicallanguage, anatural language,andapictoriallanguage[2,5].Ithasbeensuggestedthattheuseofdifferentsemioticsystems such as these supports the development of conceptual knowledge [6,7]. Furthermore, it has been recommendedthat,inanationalassessmentoflearningoutcomes[8],languagingshouldbeanintegral tothepedagogicalmethodinlearningmathematics. In our earlier paper [9], we published the outline of our pedagogical model and preliminary resultsinFinnishfromthepointofviewofintegratingschoolsubjects. Inthispaper,wewilllinkthe multimodallanguagingmodeltoamultiliteraciesframework,presentnewdata,andfurtherdiscuss theconceptofdivisionbasedonthedata. Educ.Sci.2017,7,9;doi:10.3390/educsci7010009 www.mdpi.com/journal/education Educ.Sci.2017,7,9 2of9 2. TheTheoreticalFramework 2.1. Multiliteracy The multimodal languaging model is connected to a multiliteracies framework. The basic principles of multiliteracy were set out in the manifesto of the New London Group (NLG) [10]. Sincethen,multiliteracieshasbecomeakeyconceptindescribingthechangesinthecurrenttextual world,inwhichthewrittenmodecanbecomplementedby,orevenreplacedby,othermodes[4,11]. In research, multiliteracies is usually framed by semiotics, which emphasizes the “semiotic resource” as a potential key term in meaning-making [12]. Sociocultural approaches also seem to beinherentinmultiliteracy[12]. Theseaspectsreflecttwoofthe"multis"inherentinmultiliteracies, as stated by Kalantzis and Cope [4]. First of all, the variety of modes provides more options for meaning-making;secondly,atextmayvaryenormously,dependingonsocialcontextsuchasdifferent culturalsettings,genderidentity,orthesubjectmatter[4]. Multimodal resources in school mathematics have been highlighted in several studies (e.g.,O’Halloran[13]). Theaspectofsocialdiversityofmultiliteraciesinthecontextofmathematicsis alsoacknowledged,forexample,byTakeuchi[14],whostudiedEnglishlanguagelearners(ELLs)in mathematicspracticesinanurbanCanadianclassroom. Thepresentmultimodallanguagingmodelintentionallytakesadvantageofdifferentmodesin makingmeaning. Theunderlyingideaisthat,ifthepupilsareobligedtousedifferentmodesintheir expression, theygainagreaterunderstandingofthetopic. Thus, thestudentsinthisstudywrote and drew while performing mathematical tasks. Culturally, our study is in the context of school mathematicsinFinland: thestudycombinestheinternationalsymbolicmathematicallanguagewith otherwaysofmeaning-making. Students’productionsmustthereforebeinterpretedfromthiscultural pointofview. The multiliteracies approach has only recently been incorporated into the Finnish core curriculum [15] as one of the transversal competencies. It is defined as follows [15] (p. 33): “the competence to interpret, produce and make a value judgement across a variety of different texts,whichwillhelpthepupilstounderstanddiversemodesofculturalcommunicationandtobuild theirpersonalidentity.” Wecanseethisidea, forexample, intheobjectivesofinstructionsinmathematicsforGrades 3–6 [15] (p. 398), for example: “ ... to encourage the pupil to present his or her conclusions and solutionstoothersthroughconcretetools,drawings,speech,andwriting,alsousinginformationand communicationtechnology.” Itistobenotedthatthetheoreticalconceptofmultiliteraciesisrathercomplex. Itisdescribed asbeingapedagogicalapproach,butintheFinnishcurriculumisappliedasasetofcommunication abilities [12,16]. However, there is an urgent need for research into the implementation of a multiliteracies approach in all school subjects by developing concrete yet theoretically relevant educationalmodels. 2.2. LanguagingofMathematicalThinking Traditionally, at least in Finnish mathematics education, pupils often work silently and independentlyinthemathematicsclassroom,andthesolutionstomathematicalproblemsareusually presented by mathematical symbols alone, without any clarifying text or drawing. Owing to this tradition, it is often difficult for mathematics teachers to follow how a pupil has thought through his/her solution to a mathematical problem. Has the pupil really understood the main idea of thesolution? The use of languaging breaks this tradition and serves as a means to express thinking in several different ways. It has been shown that writing and the use of natural language in the solutionsofmathematicalproblemsmayinfactboostlearninginmathematics,developmathematical understanding,changethepupil’sattitudetowardsmathematicsforthebetter,andhelptheteacher’s Educ.Sci.2017,7,9 3of9 evaluation[17]. Theuseofnaturallanguage,bothinthesolutionprocessofmathematicalproblems, Educ. Sci. 2017, 7, 9 3 of 9 andinformulatingthepresentationofthesolutions,helpsapupiltoorganizeher/hismathematical thinkingforherself/himselfandforthepeergroup[2,9,18]. Infact,itseemsthattheuseofnatural language and drawings helps most students in the solution process of mathematical problems not languageanddrawingshelpsmoststudentsinthesolutionprocessofmathematicalproblemsnotonly only at primary level, but also at higher levels of the education system [19]. atprimarylevel,butalsoathigherlevelsoftheeducationsystem[19]. At the primary school level, we have recognized three useful semiotic systems or "languages" of Attheprimaryschoollevel,wehaverecognizedthreeusefulsemioticsystemsor“languages”of meaning-making in mathematical presentations (Figure 1). These semiotic systems are a meaning-makinginmathematicalpresentations(Figure1). Thesesemioticsystemsareamathematical mathematical symbolic language, a natural language, and a pictorial language [2,12,19]. In the school symboliclanguage,anaturallanguage,andapictoriallanguage[2,12,19]. Intheschoolcontext,this context, this means students are able to express their mathematical thinking either by using meansstudentsareabletoexpresstheirmathematicalthinkingeitherbyusingmathematicalsymbols mathematical symbols (e.g., numbers, symbols), a natural language (mother tongue and/or second (e.g.,numbers,symbols),anaturallanguage(mothertongueand/orsecondlanguage),orpicturesor language), or pictures or other tangible devices [12,19]. The modes in using these are writing, othertangibledevices[12,19]. Themodesinusingthesearewriting,speaking,anddrawing. speaking, and drawing. Figure 1. The three languages that can be used to express mathematical thinking [12,20]. Figure1.Thethreelanguagesthatcanbeusedtoexpressmathematicalthinking[12,20]. Mathematical thinking is described as an information process monitored by one’s metacognition [7]. Mathematical thinking is described as an information process monitored by one’s The main purpose of using several semiotic systems in learning activities (e.g., in studying new metacognition [7]. The main purpose of using several semiotic systems in learning activities mathematical concepts and doing exercises) is to develop the student’s own meaning-making (e.g.,instudyingnewmathematicalconceptsanddoingexercises)istodevelopthestudent’sown processes. We call this process “languaging”, a concept that has been used in mathematics and in meaning-making processes. We call this process “languaging”, a concept that has been used in mother tongue didactics since the 1990s [21]. mathematicsandinmothertonguedidacticssincethe1990s[21]. Languaging in mathematics refers to expressing one’s mathematical thinking by different modes Languaginginmathematicsreferstoexpressingone’smathematicalthinkingbydifferentmodes either orally (by natural language) or in writing (by natural language, mathematical symbolic eitherorally(bynaturallanguage)orinwriting(bynaturallanguage,mathematicalsymboliclanguage, language, or pictorial language) [12,20]. From a multiliteracies aspect, languaging can be seen as a orpictoriallanguage)[12,20]. Fromamultiliteraciesaspect,languagingcanbeseenasamultimodal multimodal approach to making meanings of mathematical concepts and procedures. approachtomakingmeaningsofmathematicalconceptsandprocedures. In addition to the meaning-making of concepts and procedures for a learner, multimodal In addition to the meaning-making of concepts and procedures for a learner, multimodal languaging is also a tool for a teacher to evaluate how the learner has understood mathematics. In languaging is also a tool for a teacher to evaluate how the learner has understood mathematics. this study, we will use pupils’ texts and drawings as a resource for semantic interpretations. Inthisstudy,wewillusepupils’textsanddrawingsasaresourceforsemanticinterpretations. 2.3. The Concept of Division in School Mathematics 2.3. TheConceptofDivisioninSchoolMathematics In the Finnish National core curriculum for basic education, the objectives in division learning IntheFinnishNationalcorecurriculumforbasiceducation,theobjectivesindivisionlearning in Grades 3-6 are described as follows: “They learn division in cases of both quotition and partition. inGrades3-6aredescribedasfollows: “Theylearndivisionincasesofbothquotitionandpartition. They practise division by number units. They utilise the properties of operations and the connections Theypractisedivisionbynumberunits. Theyutilisethepropertiesofoperationsandtheconnections between them.” [15] (p. 399). betweenthem.”[15](p. 399). Pupils are introduced to the concept of division after they have learned multiplication. The objective is that they should understand the connection between multiplication and division in Grade 3. In several studies, the concept of division is observed to be difficult to understand for pupils and even for prospective teachers [22], especially in word problems and in recognizing both of the aforementioned types of division. Typically, if we ask pupils to give an example of division, they Educ.Sci.2017,7,9 4of9 Pupils are introduced to the concept of division after they have learned multiplication. The objective is that they should understand the connection between multiplication and division inGrade3. Inseveralstudies,theconceptofdivisionisobservedtobedifficulttounderstandfor pupilsandevenforprospectiveteachers[22],especiallyinwordproblemsandinrecognizingboth oftheaforementionedtypesofdivision. Typically,ifweaskpupilstogiveanexampleofdivision, theydescribeitasbeingdivisionintoparts: e.g.,“Motherhad24cookiesandshepromisedtodivide them equally to me and my five friends. How many cookies did I get?” We could describe this as the primitive model of division [22]. In fact, the model of division by contents (e.g., “For how manychildrencanmotherdividepackagesoffourcookies,ifshehas24cookies?”)isunderstoodby systemicallyteachinginuppergrades[23]. 3. TheDivisionResearch: DataCollectionandAnalysis Theuseofdifferentmodesinmathematicswasmodeledandexploredviaasix-hourteaching process, which took place in the Training School of University of Tampere in 2012 in a 4th grade classof21pupils. AllofthepupilsweremonolingualFinnishspeakers. Theprocesswasplanned and conducted in co-operation with the class teacher. After the teaching, we submitted a written questionnaireinordertoevaluatethestudents’experiencesandthoughtsontheprocess. Inthisarticle, weconcentrateonthefollowingresearchquestions: 1. Howdothepupilsinterprettheconceptofdivision? 2. Howdidthepupilsexperiencetheuseofwritinganddrawinginlearningmathematics? The key idea in the teaching process and data collection was that the students were given a mathematicalexpression(24/6−3),whichwasthenthestartingpointforseveraldifferenttasks.Atfirst, thepupilsweretoinventatypicalschoolwordproblembasedonthegivenexpression. Thismeans thattheyhadtoconstructverbalmeaningsandcontextforthenumbers,division,andsubtractionby themselves,incontrasttotheusualwaysofpresentingmathematicalproblems. Then,thestudents weretosolvetheirowninventedmathematicalwordproblembyusingthemultimodallanguaging method: inthisphase,theyworkedwithamathematicalsymboliclanguage,anaturallanguagevia writing,andapictoriallanguagebydrawing. Afterthis,theyweretoextendtheirownwordproblem intoastorybyplanningandprocessingtheirstoriesastypicallydoneindevelopingwritingskills. Thestoriesbasedonthemathematicalwordproblemwereeventuallytransformedintostripcartoon forminordertopracticetextualskillsandtoevaluatethestudent’smathematicalthinking. Aqualitativecontentanalysiswasconductedonpupils’productionsduringtheteachingprocess (wordproblems,writtenanddrawnsolutions,stories,andcartoons)andonthewrittenanswersto thequestionnaire. Basedontheproductions,wewereabletointerpretwhatkindofmeaningspupils constructedforthegivenmathematicalexpression. Theprocesswassuccessful,asnearlyallofthe pupilscooperatedwillinglyduringallthesteps. 4. Results 4.1. HowDothePupilsInterprettheConceptofDivision? Theuseofmultimodallanguagingmodelintheprocessrevealedthecontextsintowhichabstract mathematical symbolic language was referring to in the student’s thinking. Most of the students (N=21)equatedthecontextswithfood,e.g.,cookies(N=9),oranimals,e.g.,bunnies(N=6). Languagingthroughwritingrevealedhowthestudentsunderstoodthemathematicalconcepts ofdivisionandsubtraction. Intheexpression24/6−3,thekeyissueiswhatkindofmeaningspupils gave to the subtraction “minus three.” Interestingly, in the division into parts the subtraction was understoodintwodifferentways: A:“minusthree”fromonlyonegroup(N=11)orB:“minusthree” fromeachofthesixgroups(N=8)(Figure2). Educ.Sci.2017,7,9 5of9 Figure2. Thewaysthepupils(N =21)interpretedtheconceptofdivisionandsubtractioninthe expression24/6–3(modifiedfromJoutsenlahtietal.[12]). Against our expectations, we found two cases where the pupils had constructed division by contents. Thetwopupilshadproblemsinformulatingthewordproblemcorrectlytotheend,but themainideaofthedivisionbycontentswasproperlypresented(seeTable1,CaseC).Inthenext twotablesandinFigure3, weshowhowthreepupilsconstructedthewordproblem, itssolution, andthewholestoryinthecartoons. InTable1,wehavetakenthreeexamplesofthewordproblemsin whichdivisionisneededfortheirsolution. Table1.Wordproblemexamplesofdivisionsandsubtractions(TypesA,B,andCfromFigure1). TypeofDivision AnExample “Inthetreetherewere24apples.Aku,Santeri,Miina,Liisa,Kaisaand A Lauradividedthem.AfterthatKaisagavetohermother3apples. HowmanyapplesdidKaisaget?”(Pupil1) “Emmadivided24cookiestohersixfriends.Emma’ssistertook3cookies B fromeachfriend.Howmanycookiesdideachfriendget?”(Pupil2) “Samppahad24icehockeysticks.Histaskwastodividethemintogroups ofsixsticks.”Threegroupswereleftinthestorage.Howmanygroupsof C sixsticksweretakentothetraininghall?”(Thisexampleisacombination ofseveralpupils:FirsttwosentencesarefromPupil3andtherestofthe problemfromothers) In the second phase, pupils constructed solutions to their word problem firstly via natural language,andsecondlyviapictoriallanguage(Table2). Thesolutionsbynaturallanguagewerelike littlestories,whichalsocontainedmathematicalsymboliclanguage. Theoriginalhandmadedrawings were completed as shown in Table 2 (drawn by computer). Only Pupil 3 of the three pupils had problemsinlogicallymakingasolutionandcartoons. Pupil3hadmadethebeginningofthesolution ofthewordproblemcorrectly,buttherestofthesolutionwasinsufficient. Wepresenthereonlythe correctpartsoftheexampleofModelC. 1 Educ.Sci.2017,7,9 6of9 Table2.Pupils’wordproblemsolutionsofdivisionsandsubtractionsbynaturalandpictoriallanguage (TypesA,B,andCfromFigure1).Thesolutionsviapictoriallanguagearebasedonpupils’drawings. TypeofDivision TheSolutionbyNaturalandPictorialLanguage EEdduucc.. SSccii.. 22001177,, 77,, 99 “Kaisagotatfirst24/6=4(apples).WhenKaisacamehomeshegave 66 ooff 99 Educ. Sci. 2017, 7, 9 6 of 9 threeapplestothemother4−3=1.Kaisagotoneapple.”(Pupil1). A “24/6 = 4, 4 – 3 = 1. The cookies were divided and everyone got four “““222444///666 === 444,, ,444 ––– 333 === 111.. .TTThhheee ccoocooookkkiieeiess swwweeerreere dddiivviviiddideeeddd aaannnddd eeevvveeerrryyyooonnneee gggooott tffoofouuurrr (cookies). The sister takes from everyone else three cookies. Every girl (((cccooooookkkiiieeesss)))... TTThhheee sssiiisssttteeerrr tttaaakkkeeesss f ffrrrooommme eevvveeerrryyyooonnneeee eelsllseseet htthhrerreeeeec occoooookkkieiisee.ss.E. EEvvvereeyrryyg gigriilrrgll ot gogonotlt y oononlnlyye oocnnoeoe kccoioeoo.”kkii(eeP..”u” p((PPiluu2pp)i.ill 22)).. B got only one cookie.” (Pupil 2). BBB ““2244//66 −− 33 == 11.. SSaammppppaa ddiivviiddeess tthhee iiccee hhoocckkeeyy ssttiiccss iinnttoo ssiixx ggrroouuppss.. HHee l“l“ee22ff4t4t / t/th66h rr−−ee e3e 3 g=g=r r1oo.1u uS.ppaSsmsa miinpnp p tthpahea ed sdistvtiovoirdiradaegegses et htaahnene didci c tetoeo hoohokkoc cokokneneeyey gsgstrtrioioccusus pipinn iitntnoocc slsluiuixxdd giginnrrgoog uu ffoppoussu..r r HH iicceeee l eft hthlheoofrcctek ketehegyyrr e sosettu iigccpkrksossu i tntopo ts tht hhienee s tttthrroaaeri iansngtiionenrggaa ncgcadaem matopnpod..”k” t (o(oTTnowwkeo ogo rsnsoeeeunn gptterenoinnucccepelsus i fdnfrricoonlmumgd f PPoinuuugppr ifiilolc 3e3u arha nonicdcdek ey shtoicckkseyto stthicektsr atoin tihneg tcraaminpin.”g (cTawmops.”e n(Ttewnoce ssefnrtoemncPeus pfriolm3a Pnudptihl e3 raensdt of CCCC tttthhhheeee rsrreoeesssltutt otooifoff tntthhhefeer sossomoollluuuotttitiiohoonenn r fsffrrr)oo.ommm ooottthhheeerrrsss)))... AAtt tthhee eenndd,, ppuuppiillss ddrreeww ccaarrttoooonnss iinn wwhhiicchh tthheeyy ttoolldd tthhee wwhhoollee ssttoorryy:: tthhee wwoorrdd pprroobblleemm aanndd iittss At the end, pupils drew cartoons in which they told the whole story: the word problem and its ssoolluuttiiAoonnt t((hFFeiiggeuunrrdee , 33p)).u. PpPuiulpspiidll r11e wddrrceeawwr t tothhoeen cscaairnrttoowoohnni caahbboothuuett y hhotoowwld KKthaaieissawa dhdioivvlieiddseetddo r 22y44: atahppeppwlleesos,,r dPPuuppprioillb 22l e ddmrreeawwn d tthhietes solution (Figure 3). Pupil 1 drew the cartoon about how Kaisa divided 24 apples, Pupil 2 drew the ccsaaorrlttuoootoinon n aabb(oFouiugttu hhrooeww3 ) EE.mmPummpaai l dd1iivvdiidrdeeewdd 22th44 e ccoocoaokrktiioeeoss n ((CCaaabssoeeu BBt iihnno FFwiiggKuurareeis 11a)),,d aainvndidd PPeudupp2ii4ll 33a dpdrpreelwews ,tthhPeeu ccpaairlrtt2oooodnnr e aawbbootuuhtte cartoon about how Emma divided 24 cookies (Case B in Figure 1), and Pupil 3 drew the cartoon about hhcooawwrt o SSoaanmmappbppoaua tddhiivoviwiddeeEddm 22m44 aiiccdee i hvhoiodccekkdeeyy2 4ssttciiocckoksks i((eCCsaa(sCseea CsCe iiBnn iFFniiggFuuigrreue r11e)).1. T)T,hhaeen ldlaasPsttu ccpaairrltt3oooodnnr,e, wCC,, t ihisse nncooattr tccoooommnppallbeeotteeu,, t how Samppa divided 24 ice hockey sticks (Case C in Figure 1). The last cartoon, C, is not complete, bbheeoccawauusSseae mtthhpeerpreea wwdeievrreied sseoodmm2ee4 mmiciiessuuhnnoddckeererssyttaasnntiddcikinnsgg(ssC aaatts tethhCee eeinnnddF ioogffu ttrhheee 1 ss)oo.llTuuhttiieoonnla.. s t cartoon,C,isnotcomplete, because there were some misunderstandings at the end of the solution. becausethereweresomemisunderstandingsattheendofthesolution. FFiigguurree 33.. TThhrreeee eexxaammpplleess ooff ppuuppiillss’’ ccaarrttoooonnss ((MMooddeellss AA,, BB,, aanndd CC ffrroomm FFiigguurree 11)).. TThhee ssppeeeecchh bbuubbbblleess Figure 3. Three examples of pupils’ cartoons (Models A, B, and C from Figure 1). The speech bubbles hhaavvee bbeeeenn ttrraannssllaatteedd ffrroomm FFiinnnniisshh ttoo EEnngglliisshh.. have been translated from Finnish to English. Figure3.Threeexamplesofpupils’cartoons(ModelsA,B,andCfromFigure1).Thespeechbubbles AAAhlalllllv tttehhhbeeee ppepnuuuptppriiailllsnss s iiilnnna t tetthhhdeeef rsssotttumuudddFyyyi n sssnuuuicsccchcceeeteoeedddEeeendddg liiiinnnsh ddd.rrraaawwwiiinnnggg lllooogggiiicccaaalll cccaaarrrtttoooooonnnsss fffrrrooommm ttthhheeeiiirrr wwwooorrrddd ppprrrooobbbllleeemmm... AAAllllll tthhrreeee ppuuppiillss ccrreeaatteedd aa rreelliiaabbllee ccoonntteexxtt iinn tthheeiirr wwoorrdd pprroobblleemm ((TTaabbllee 11)) aanndd iinn tthheeiirr ccaarrttoooonnss ((FFiigguurree 33)) ffoorr three pupils created a reliable context in their word problem (Table 1) and in their cartoons (Figure 3) for ttthhheee mmmaAaattthlhhleeetmmmheaaatttpiiicccuaaaplll eieelxxxspppirnrreeessstshssiiieooonnns.t.. uTTTdhhheyee dddsiuiivvvciiiscssieiioooennnd ieiinnndtttoooi n pppaadarrrrtttasss w aaannnindddg ttthhhloeee g sssuiuucbbabtlttrrracaaaccctrttitiioooonnno n hhhsaaavvvfereeo aaamcccqqqutuuhiiirerreieerddd w rrreeeoaaarlll d mmmpeeearaaonnnbiiinnlnegggmsss . iiinAnn ltttlhhheteeh ppprueuuepppiiiplll’’’usss p oooiwwwlsnnnc crccaeaararrttttoooeodoonnnasss fffrrrreooolmimmab tttlhhheeeeiciirrro ppnpootoeiiinxnntttt oooinfff vvvtihiieeeewwwirsss...w ord problem (Table 1) and in their cartoons (Figure3)forthemathematicalexpression. Thedivisionintopartsandthesubtractionhaveacquired 444r...e222a... lHHHmooowewwa DnDDiiiinddd g tttshhheeie n PPPutuuhpppeiiilllspss uEEEpxxxpippleee’rsrriiieoeennnwcccenee tttchhhaeeer UUtUosssoeeen ooosfff WfWWrorrrmiiitttiiinnnthggg e aaainrnndddp DoDDirnrraaatwwwoiiifnnnvggg i ieiinnnw MMMs.aaattthhheeemmmaaatttiiicccsss LLLeeeaaarrrnnniiinnnggg??? AAfftteerr tthhee tteeaacchhiinngg pprroocceessss,, wwee aasskkeedd tthhee ppuuppiillss hhooww tthheeyy hhaadd eexxppeerriieenncceedd tthhee mmuullttiimmooddaall After the teaching process, we asked the pupils how they had experienced the multimodal aapppprrooaacchh iinn mmaatthheemmaattiiccss lleeaarrnniinngg bbyy aa qquueessttiioonnnnaaiirree.. TThhee rreessuullttss aarree sshhoowwnn iinn TTaabblleess 33––55.. MMoosstt ooff approach in mathematics learning by a questionnaire. The results are shown in Tables 3–5. Most of tthhee ppuuppiillss lliikkeedd wwrriittiinngg tthhee wwoorrdd pprroobblleemm aanndd ccoonnssttrruuccttiinngg aa llaarrggeerr ssttoorryy aabboouutt iitt.. AAnn iinntteerreessttiinngg the pupils liked writing the word problem and constructing a larger story about it. An interesting ddeettaaiill iinn TTaabbllee 33 iiss tthhaatt aallmmoosstt aallll ooff tthhee ggiirrllss lliikkeedd wwrriittiinngg tthhee mmoosstt,, wwhheerreeaass mmoosstt ooff tthhee bbooyyss lliikkeedd detail in Table 3 is that almost all of the girls liked writing the most, whereas most of the boys liked ddrraawwiinngg ccaarrttoooonnss.. AA mmiinndd mmaapp wwaass nneeeeddeedd iinn tthhee ppllaannnniinngg ooff tthhee ssttoorryy;; tthhaatt ppaarrtt ooff tthhee ssttuuddyy iiss drawing cartoons. A mind map was needed in the planning of the story; that part of the study is oommiitttteedd ffrroomm tthhiiss aarrttiiccllee.. MMiinndd mmaapp mmaakkiinngg ddeevveellooppss aannaallyyttiiccaall tthhiinnkkiinngg.. omitted from this article. Mind map making develops analytical thinking. 1 Educ.Sci.2017,7,9 7of9 4.2. HowDidthePupilsExperiencetheUseofWritingandDrawinginMathematicsLearning? After the teaching process, we asked the pupils how they had experienced the multimodal approachinmathematicslearningbyaquestionnaire. TheresultsareshowninTables3–5. Mostofthe pupilslikedwritingthewordproblemandconstructingalargerstoryaboutit. Aninterestingdetailin Table3isthatalmostallofthegirlslikedwritingthemost,whereasmostoftheboyslikeddrawing cartoons. Amindmapwasneededintheplanningofthestory;thatpartofthestudyisomittedfrom thisarticle. Mindmapmakingdevelopsanalyticalthinking. Table3.Themostagreeablepartintheprojectfromthepupils’pointsofview(N=21). Girls Boys Drawingcartoons 1 6 Writingwordproblemandstory 11 1 Makingmindmapforastory 0 2 Thepupilswereaskedwhethertheythoughtthatwriting(theuseofnaturallanguage)supported thesolutionmakingforwordproblems. Almostallofthepupilsthoughtthatwritingsupportedit (Table4). Theresultisinteresting,becausewritingwasnotfavoredbytheboys. Table 4. Do you think that writing (natural language) supports the solutions of word problems? (N=21). Answer Girls Boys Yes 12 6 No 0 2 Noanswer 0 1 The pupils also provided their opinions as to why writing is important or not in the solving process. By content analysis, four main themes were found, which are presented in Table 5 with textexamples. Table5. Examplesofpupils'opinionsontheimportanceofusingnaturallanguageinsolutionsof wordproblems. Theme Examples Betterunderstanding “IunderstandbetterhowIhavesolvedtheproblem.”(5pupils) Easiertoconstructthesolution “Itiseasiertosolvetheproblemwhenyoucanwrite.”(5pupils) Checkingoftheanswer “Youcanjustifyfromthewrittentextthatyouhavesolveditcorrectly”(2pupils) Mothertonguelearning(writing) “Youlearnatsametimemathematicsandmothertongue”(1pupil) 5. Discussion The pedagogical model described in this paper is an example of using multimodal ways of meaning-makinginschoolmathematics.Theintentionaluseofmultimodallanguagingmodelrevealed themeaningspupilsmadeforthemathematicalsymboliclanguageandconcepts(here,divisionand subtraction)andservedininterpretingtheirthinking. Thesevarioustypesofwaystoexpressone’s thinkingservealsoasawayfortheteachertoevaluatethestudents’understandingoftheconcepts. Despitetheunfamiliarityofthetask,everypupilintheclassmanagedtoproduceaproperword problem,solutionsbynaturallanguage,andapictoriallanguagefortheproblemandthecartoons. Themosttypicalinterpretationofdivisionwasdivisionintoparts(Figure1). Fromthepointofviewof thepupils,writingwasseenasusefulforabetterunderstandingofthesolution,andforconstructing thesolutionmoreeasily(Tables4and5). Asimilarresulthasalsobeenobservedinthelanguaging Educ.Sci.2017,7,9 8of9 surveysofuniversitymathematicsteaching. Thegirlslikedwritingmorethantheboys,andtheboys likeddrawingmorethanthegirls(Table3). Boyshavingproblemsinwritingareseenonawiderscale inFinland;nationalFinnishassessmentsoflearningoutcomesattheendofcomprehensiveschool haveshownthatboysaresignificantlyworsethangirlsinwriting[24]. Therewerenodifferencesin thedatabetweenhowpupilsmadesolutionsvianaturalorpictoriallanguages: allpupilscoulddo bothofthemmainlycorrectly. Allpupilswereabletodrawcartoons,presentingboththeproblemand itssolution. Wesuggestthatthebroadenedwaysofexpressingmathematicalthinkingmayhelpthosewho struggle with mathematics and for whom mathematical symbolic language as such is difficult to comprehend. Theuseofwritinganddrawinginproblemsolvingmayalsostrengthenthelearningof mathematicalconcepts,astheuseofdifferentmodesleadstoorganizingone’smathematicalthinking. Inthisstudy,allofthepupilsweremonolingualFinnishspeakers;however,theuseofmultimodality mayalsosupport,forexample,L2learnersforwhomthepictoriallanguagemayserveasawayto understandmathematicalconcepts. Theuseofdifferentmodesinlearningmathematicscouldeasilybeextendedtotheuseof,for example,videos. Theuseofvariousmodesandevendigitaltechnologywouldconnectthesymbolic mathematicallanguagetomorefamiliarwaysofmeaning-makingforyoungstudents. Inabroader sense,thesekindsofeducationalapplicationsarecloselyrelatedtoamultiliteraciesapproachand serveasawaytounderstandthevariouswaysofmakingmeaningintodaystextualenvironment. Afewlimitationsofthisstudyaretobementioned. Firstofall,thedatawascollectedinaspecific educationalcontext(monolingualFinnishclassroom);secondly,thesamplesizeissmall,asiscommon incasestudies. Nevertheless,thedatashowedpotentialindevelopingthemultimodallanguaging model,asitservedtoexpressthinking. Themultiliteraciesframeworkwithmultimodalitybringsnewinsightsintoschoolmathematics. Furtherresearchisneededintohowtheintentionaluseofdifferentmodesservesinlearningspecific concepts,suchasmultiplication,orinmultilingualclassrooms. Acknowledgments: WewouldliketothankclassteacherMarjaTuomiforco-operationandforprovidingher classforthisstudy. Author Contributions: Both authors contributed to the development of this paper. Jorma Joutsenlahti and PirjoKuljuconcievedanddesignedthedatacollection.JormaJoutsenlahtianalysedthedataregardingtheconcept ofdivision.Bothauthorsparticipatedtothewritingofthispaper. ConflictsofInterest:Theauthorsdeclarenoconflictofinterest. References 1. Bauersfeld,H.Languagegamesinthemathematicsclassroom: Theirfunctionandtheireffects. 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