ERIC EJ1135084: Multimodal Languaging as a Pedagogical Model--A Case Study of the Concept of Division in School Mathematics

2017·4 MB·English

by  ERIC

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.:[phone]
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.
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