educa ion
sciences
A icle
Mul imodal Languaging as a Pedagogical Model—A
Case S udy o he Concep o Di ision in
School Ma hema ics
Jo ma Jou senlah i * and Pi jo Kulju
School o Educa ion, Uni e si y o Tampe e, Tampe e 33100, Finland; [email p o ec ed]
*Co espondence: [email p o ec ed]; Tel.: +358-505-439-197
Academic Edi o : Eila Je onen
Recei ed: 31 Oc obe 2016; Accep ed: 28 Decembe 2016; Published: 4 Janua y 2017
Abs ac :
The pu pose o his s udy is o p esen a mul imodal languaging model o ma hema ics
educa ion. The model consis s o ma hema ical symbolic language, a pic o ial language, and a
na u al language. By applying his model, he objec i e was o s udy how 4 h g ade pupils (N= 21)
unde s and he concep o di ision. The da a was collec ed o e six hou s o eaching sessions,
du ing which he pupils exp essed hei ma hema ical hinking mainly by w i ing and d awing.
Thei p oduc ions, as well as ques ionnai e a e he p ocess, we e analyzed quali a i ely. The esul s
show ha , in exp essing he ma hema ical p oblem in e bal o m, mos o he s uden s saw i as
a di ision in o pa s. I was e iden om he pupils’ ex s and d awings ha he ma hema ical
exp ession o sub ac ion could be in e p e ed in h ee di e en ways. I was ound ha he pupils
enjoyed using w i ing in he solu ion o wo d p oblems, and i is sugges ed ha he use o di e en
modes in exp essing ma hema ical hinking may bo h s eng hen he lea ning o ma hema ical
concep s and suppo he e alua ion o lea ning.
Keywo ds: mul ili e acy; languaging; di ision; ma hema ics; educa ion
1. In oduc ion
T adi ionally, he e has been a s ong emphasis on he use o symbolic ma hema ical language
in ep esen ing ma hema ics. This has, howe e , been ound o be a limi ing ac o in exp essing
ma hema ical hinking in lea ning p ocesses [
1
–
3
]. The aim o his a icle is o p esen a mul imodal
languaging model, in which he ways o exp ess ma hema ical hinking a e expanded beyond
ma hema ic symbolic language. A second objec i e is o obse e how 4 h g ade pupils unde s and he
concep o di ision based on his model.
Di e en ways o exp essing hinking and making meaning o m he unde lying heo e ical basis
o his s udy. Theo e ically, he mul imodal languaging model is ela ed o mul ili e acy, a concep
e e ing o a ious modes in he cu en communica ions en i onmen [
4
]. The p esen model includes
h ee ypes o semio ic sys ems o meaning-making: a symbolic ma hema ical language, a na u al
language, and a pic o ial language [
2
,
5
]. I has been sugges ed ha he use o di e en semio ic sys ems
such as hese suppo s he de elopmen o concep ual knowledge [
6
,
7
]. Fu he mo e, i has been
ecommended ha , in a na ional assessmen o lea ning ou comes [
8
], languaging should be an in eg al
o he pedagogical me hod in lea ning ma hema ics.
In ou ea lie pape [
9
], we published he ou line o ou pedagogical model and p elimina y
esul s in Finnish om he poin o iew o in eg a ing school subjec s. In his pape , we will link he
mul imodal languaging model o a mul ili e acies amewo k, p esen new da a, and u he discuss
he concep o di ision based on he da a.
Educ. Sci. 2017,7, 9; doi:10.3390/educsci7010009 www.mdpi.com/jou nal/educa ion
Educ. Sci. 2017,7, 9 2 o 9
2. The Theo e ical F amewo k
2.1. Mul ili e acy
The mul imodal languaging model is connec ed o a mul ili e acies amewo k. The basic
p inciples o mul ili e acy we e se ou in he mani es o o he New London G oup (NLG) [
10
].
Since hen, mul ili e acies has become a key concep in desc ibing he changes in he cu en ex ual
wo ld, in which he w i en mode can be complemen ed by, o e en eplaced by, o he modes [4,11].
In esea ch, mul ili e acies is usually amed by semio ics, which emphasizes he “semio ic
esou ce” as a po en ial key e m in meaning-making [
12
]. Sociocul u al app oaches also seem o
be inhe en in mul ili e acy [
12
]. These aspec s e lec wo o he "mul is" inhe en in mul ili e acies,
as s a ed by Kalan zis and Cope [
4
]. Fi s o all, he a ie y o modes p o ides mo e op ions o
meaning-making; secondly, a ex may a y eno mously, depending on social con ex such as di e en
cul u al se ings, gende iden i y, o he subjec ma e [4].
Mul imodal esou ces in school ma hema ics ha e been highligh ed in se e al s udies
(e.g., O’Hallo an [
13
]). The aspec o social di e si y o mul ili e acies in he con ex o ma hema ics is
also acknowledged, o example, by Takeuchi [
14
], who s udied English language lea ne s (ELLs) in
ma hema ics p ac ices in an u ban Canadian class oom.
The p esen mul imodal languaging model in en ionally akes ad an age o di e en modes in
making meaning. The unde lying idea is ha , i he pupils a e obliged o use di e en modes in hei
exp ession, hey gain a g ea e unde s anding o he opic. Thus, he s uden s in his s udy w o e
and d ew while pe o ming ma hema ical asks. Cul u ally, ou s udy is in he con ex o school
ma hema ics in Finland: he s udy combines he in e na ional symbolic ma hema ical language wi h
o he ways o meaning-making. S uden s’ p oduc ions mus he e o e be in e p e ed om his cul u al
poin o iew.
The mul ili e acies app oach has only ecen ly been inco po a ed in o he Finnish co e
cu iculum [
15
] as one o he ans e sal compe encies. I is de ined as ollows [
15
] (p. 33):
“ he compe ence o in e p e , p oduce and make a alue judgemen ac oss a a ie y o di e en
ex s, which will help he pupils o unde s and di e se modes o cul u al communica ion and o build
hei pe sonal iden i y.”
We can see his idea, o example, in he objec i es o ins uc ions in ma hema ics o G ades
3–6 [
15
] (p. 398), o example: “
. . .
o encou age he pupil o p esen his o he conclusions and
solu ions o o he s h ough conc e e ools, d awings, speech, and w i ing, also using in o ma ion and
communica ion echnology.”
I is o be no ed ha he heo e ical concep o mul ili e acies is a he complex. I is desc ibed
as being a pedagogical app oach, bu in he Finnish cu iculum is applied as a se o communica ion
abili ies [
12
,
16
]. Howe e , he e is an u gen need o esea ch in o he implemen a ion o a
mul ili e acies app oach in all school subjec s by de eloping conc e e ye heo e ically ele an
educa ional models.
2.2. Languaging o Ma hema ical Thinking
T adi ionally, a leas in Finnish ma hema ics educa ion, pupils o en wo k silen ly and
independen ly in he ma hema ics class oom, and he solu ions o ma hema ical p oblems a e usually
p esen ed by ma hema ical symbols alone, wi hou any cla i ying ex o d awing. Owing o his
adi ion, i is o en di icul o ma hema ics eache s o ollow how a pupil has hough h ough
his/he solu ion o a ma hema ical p oblem. Has he pupil eally unde s ood he main idea o
he solu ion?
The use o languaging b eaks his adi ion and se es as a means o exp ess hinking in
se e al di e en ways. I has been shown ha w i ing and he use o na u al language in he
solu ions o ma hema ical p oblems may in ac boos lea ning in ma hema ics, de elop ma hema ical
unde s anding, change he pupil’s a i ude owa ds ma hema ics o he be e , and help he eache ’s
Educ. Sci. 2017,7, 9 3 o 9
e alua ion [
17
]. The use o na u al language, bo h in he solu ion p ocess o ma hema ical p oblems,
and in o mula ing he p esen a ion o he solu ions, helps a pupil o o ganize he /his ma hema ical
hinking o he sel /himsel and o he pee g oup [
2
,
9
,
18
]. In ac , i seems ha he use o na u al
language and d awings helps mos s uden s in he solu ion p ocess o ma hema ical p oblems no only
a p ima y le el, bu also a highe le els o he educa ion sys em [19].
A he p ima y school le el, we ha e ecognized h ee use ul semio ic sys ems o “languages” o
meaning-making in ma hema ical p esen a ions (Figu e 1). These semio ic sys ems a e a ma hema ical
symbolic language, a na u al language, and a pic o ial language [
2
,
12
,
19
]. In he school con ex , his
means s uden s a e able o exp ess hei ma hema ical hinking ei he by using ma hema ical symbols
(e.g., numbe s, symbols), a na u al language (mo he ongue and/o second language), o pic u es o
o he angible de ices [12,19]. The modes in using hese a e w i ing, speaking, and d awing.
Educ. Sci. 2017, 7, 9 3 o 9
language and d awings helps mos s uden s in he solu ion p ocess o ma hema ical p oblems no
only a p ima y le el, bu also a highe le els o he educa ion sys em [19].
A he p ima y school le el, we ha e ecognized h ee use ul semio ic sys ems o "languages" o
meaning-making in ma hema ical p esen a ions (Figu e 1). These semio ic sys ems a e a
ma hema ical symbolic language, a na u al language, and a pic o ial language [2,12,19]. In he school
con ex , his means s uden s a e able o exp ess hei ma hema ical hinking ei he by using
ma hema ical symbols (e.g., numbe s, symbols), a na u al language (mo he ongue and/o second
language), o pic u es o o he angible de ices [12,19]. The modes in using hese a e w i ing,
speaking, and d awing.
Figu e 1. The h ee languages ha can be used o exp ess ma hema ical hinking [12,20].
Ma hema ical hinking is desc ibed as an in o ma ion p ocess moni o ed by one’s me acogni ion [7].
The main pu pose o using se e al semio ic sys ems in lea ning ac i i ies (e.g., in s udying new
ma hema ical concep s and doing exe cises) is o de elop he s uden ’s own meaning-making
p ocesses. We call his p ocess “languaging”, a concep ha has been used in ma hema ics and in
mo he ongue didac ics since he 1990s [21].
Languaging in ma hema ics e e s o exp essing one’s ma hema ical hinking by di e en modes
ei he o ally (by na u al language) o in w i ing (by na u al language, ma hema ical symbolic
language, o pic o ial language) [12,20]. F om a mul ili e acies aspec , languaging can be seen as a
mul imodal app oach o making meanings o ma hema ical concep s and p ocedu es.
In addi ion o he meaning-making o concep s and p ocedu es o a lea ne , mul imodal
languaging is also a ool o a eache o e alua e how he lea ne has unde s ood ma hema ics. In
his s udy, we will use pupils’ ex s and d awings as a esou ce o seman ic in e p e a ions.
2.3. The Concep o Di ision in School Ma hema ics
In he Finnish Na ional co e cu iculum o basic educa ion, he objec i es in di ision lea ning
in G ades 3-6 a e desc ibed as ollows: “They lea n di ision in cases o bo h quo i ion and pa i ion.
They p ac ise di ision by numbe uni s. They u ilise he p ope ies o ope a ions and he connec ions
be ween hem.” [15] (p. 399).
Pupils a e in oduced o he concep o di ision a e hey ha e lea ned mul iplica ion. The
objec i e is ha hey should unde s and he connec ion be ween mul iplica ion and di ision in G ade
3. In se e al s udies, he concep o di ision is obse ed o be di icul o unde s and o pupils and
e en o p ospec i e eache s [22], especially in wo d p oblems and in ecognizing bo h o he
a o emen ioned ypes o di ision. Typically, i we ask pupils o gi e an example o di ision, hey
Figu e 1. The h ee languages ha can be used o exp ess ma hema ical hinking [12,20].
Ma hema ical hinking is desc ibed as an in o ma ion p ocess moni o ed by one’s
me acogni ion [
7
]. The main pu pose o using se e al semio ic sys ems in lea ning ac i i ies
(e.g., in s udying new ma hema ical concep s and doing exe cises) is o de elop he s uden ’s own
meaning-making p ocesses. We call his p ocess “languaging”, a concep ha has been used in
ma hema ics and in mo he ongue didac ics since he 1990s [21].
Languaging in ma hema ics e e s o exp essing one’s ma hema ical hinking by di e en modes
ei he o ally (by na u al language) o in w i ing (by na u al language, ma hema ical symbolic language,
o pic o ial language) [
12
,
20
]. F om a mul ili e acies aspec , languaging can be seen as a mul imodal
app oach o making meanings o ma hema ical concep s and p ocedu es.
In addi ion o he meaning-making o concep s and p ocedu es o a lea ne , mul imodal
languaging is also a ool o a eache o e alua e how he lea ne has unde s ood ma hema ics.
In his s udy, we will use pupils’ ex s and d awings as a esou ce o seman ic in e p e a ions.
2.3. The Concep o Di ision in School Ma hema ics
In he Finnish Na ional co e cu iculum o basic educa ion, he objec i es in di ision lea ning
in G ades 3-6 a e desc ibed as ollows: “They lea n di ision in cases o bo h quo i ion and pa i ion.
They p ac ise di ision by numbe uni s. They u ilise he p ope ies o ope a ions and he connec ions
be ween hem.” [15] (p. 399).
Educ. Sci. 2017,7, 9 4 o 9
Pupils a e in oduced o he concep o di ision a e hey ha e lea ned mul iplica ion.
The objec i e is ha hey should unde s and he connec ion be ween mul iplica ion and di ision
in G ade 3. In se e al s udies, he concep o di ision is obse ed o be di icul o unde s and o
pupils and e en o p ospec i e eache s [
22
], especially in wo d p oblems and in ecognizing bo h
o he a o emen ioned ypes o di ision. Typically, i we ask pupils o gi e an example o di ision,
hey desc ibe i as being di ision in o pa s: e.g., “Mo he had 24 cookies and she p omised o di ide
hem equally o me and my i e iends. How many cookies did I ge ?” We could desc ibe his
as he p imi i e model o di ision [
22
]. In ac , he model o di ision by con en s (e.g., “Fo how
many child en can mo he di ide packages o ou cookies, i she has 24 cookies?”) is unde s ood by
sys emically eaching in uppe g ades [23].
3. The Di ision Resea ch: Da a Collec ion and Analysis
The use o di e en modes in ma hema ics was modeled and explo ed ia a six-hou eaching
p ocess, which ook place in he T aining School o Uni e si y o Tampe e in 2012 in a 4 h g ade
class o 21 pupils. All o he pupils we e monolingual Finnish speake s. The p ocess was planned
and conduc ed in co-ope a ion wi h he class eache . A e he eaching, we submi ed a w i en
ques ionnai e in o de o e alua e he s uden s’ expe iences and hough s on he p ocess. In his a icle,
we concen a e on he ollowing esea ch ques ions:
1. How do he pupils in e p e he concep o di ision?
2. How did he pupils expe ience he use o w i ing and d awing in lea ning ma hema ics?
The key idea in he eaching p ocess and da a collec ion was ha he s uden s we e gi en a
ma hema ical exp ession (24/6
−
3), which was hen he s a ing poin o se e al di e en asks. A i s ,
he pupils we e o in en a ypical school wo d p oblem based on he gi en exp ession. This means
ha hey had o cons uc e bal meanings and con ex o he numbe s, di ision, and sub ac ion by
hemsel es, in con as o he usual ways o p esen ing ma hema ical p oblems. Then, he s uden s
we e o sol e hei own in en ed ma hema ical wo d p oblem by using he mul imodal languaging
me hod: in his phase, hey wo ked wi h a ma hema ical symbolic language, a na u al language ia
w i ing, and a pic o ial language by d awing. A e his, hey we e o ex end hei own wo d p oblem
in o a s o y by planning and p ocessing hei s o ies as ypically done in de eloping w i ing skills.
The s o ies based on he ma hema ical wo d p oblem we e e en ually ans o med in o s ip ca oon
o m in o de o p ac ice ex ual skills and o e alua e he s uden ’s ma hema ical hinking.
A quali a i e con en analysis was conduc ed on pupils’ p oduc ions du ing he eaching p ocess
(wo d p oblems, w i en and d awn solu ions, s o ies, and ca oons) and on he w i en answe s o
he ques ionnai e. Based on he p oduc ions, we we e able o in e p e wha kind o meanings pupils
cons uc ed o he gi en ma hema ical exp ession. The p ocess was success ul, as nea ly all o he
pupils coope a ed willingly du ing all he s eps.
4. Resul s
4.1. How Do he Pupils In e p e he Concep o Di ision?
The use o mul imodal languaging model in he p ocess e ealed he con ex s in o which abs ac
ma hema ical symbolic language was e e ing o in he s uden ’s hinking. Mos o he s uden s
(N= 21) equa ed he con ex s wi h ood, e.g., cookies (N= 9), o animals, e.g., bunnies (N= 6).
Languaging h ough w i ing e ealed how he s uden s unde s ood he ma hema ical concep s
o di ision and sub ac ion. In he exp ession 24/6
−
3, he key issue is wha kind o meanings pupils
ga e o he sub ac ion “minus h ee.” In e es ingly, in he di ision in o pa s he sub ac ion was
unde s ood in wo di e en ways: A: “minus h ee” om only one g oup (N= 11) o B: “minus h ee”
om each o he six g oups (N= 8) (Figu e 2).
Educ. Sci. 2017,7, 9 5 o 9
1
Figu e 2.
The ways he pupils (N= 21) in e p e ed he concep o di ision and sub ac ion in he
exp ession 24/6–3 (modi ied om Jou senlah i e al. [12]).
Agains ou expec a ions, we ound wo cases whe e he pupils had cons uc ed di ision by
con en s. The wo pupils had p oblems in o mula ing he wo d p oblem co ec ly o he end, bu
he main idea o he di ision by con en s was p ope ly p esen ed (see Table 1, Case C). In he nex
wo ables and in Figu e 3, we show how h ee pupils cons uc ed he wo d p oblem, i s solu ion,
and he whole s o y in he ca oons. In Table 1, we ha e aken h ee examples o he wo d p oblems in
which di ision is needed o hei solu ion.
Table 1. Wo d p oblem examples o di isions and sub ac ions (Types A, B, and C om Figu e 1).
Type o Di ision An Example
A
“In he ee he e we e 24 apples. Aku, San e i, Miina, Liisa, Kaisa and
Lau a di ided hem. A e ha Kaisa ga e o he mo he 3 apples.
How many apples did Kaisa ge ?” (Pupil 1)
B
“Emma di ided 24 cookies o he six iends. Emma’s sis e ook 3 cookies
om each iend. How many cookies did each iend ge ?” (Pupil 2)
C
“Samppa had 24 ice hockey s icks. His ask was o di ide hem in o g oups
o six s icks.” Th ee g oups we e le in he s o age. How many g oups o
six s icks we e aken o he aining hall?” (This example is a combina ion
o se e al pupils: Fi s wo sen ences a e om Pupil 3 and he es o he
p oblem om o he s)
In he second phase, pupils cons uc ed solu ions o hei wo d p oblem i s ly ia na u al
language, and secondly ia pic o ial language (Table 2). The solu ions by na u al language we e like
li le s o ies, which also con ained ma hema ical symbolic language. The o iginal handmade d awings
we e comple ed as shown in Table 2(d awn by compu e ). Only Pupil 3 o he h ee pupils had
p oblems in logically making a solu ion and ca oons. Pupil 3 had made he beginning o he solu ion
o he wo d p oblem co ec ly, bu he es o he solu ion was insu icien . We p esen he e only he
co ec pa s o he example o Model C.

Educ. Sci. 2017,7, 9 6 o 9
Table 2.
Pupils’ wo d p oblem solu ions o di isions and sub ac ions by na u al and pic o ial language
(Types A, B, and C om Figu e 1). The solu ions ia pic o ial language a e based on pupils’ d awings.
Type o Di ision The Solu ion by Na u al and Pic o ial Language
A
“Kaisa go a i s 24/6 = 4 (apples). When Kaisa came home she ga e
h ee apples o he mo he 4 −3 = 1. Kaisa go one apple.” (Pupil 1).
Educ. Sci. 2017, 7, 9 6 o 9
B
“24/6 = 4, 4 – 3 = 1. The cookies we e di ided and e e yone go ou
(cookies). The sis e akes om e e yone else h ee cookies. E e y gi l
go only one cookie.” (Pupil 2).
C
“24/6 − 3 = 1. Samppa di ides he ice hockey s ics in o six g oups. He
le h ee g oups in he s o age and ook one g oup including ou ice
hockey s icks o he aining camp.” (Two sen ences om Pupil 3 and
he es o he solu ion om o he s).
A he end, pupils d ew ca oons in which hey old he whole s o y: he wo d p oblem and i s
solu ion (Figu e 3). Pupil 1 d ew he ca oon abou how Kaisa di ided 24 apples, Pupil 2 d ew he
ca oon abou how Emma di ided 24 cookies (Case B in Figu e 1), and Pupil 3 d ew he ca oon abou
how Samppa di ided 24 ice hockey s icks (Case C in Figu e 1). The las ca oon, C, is no comple e,
because he e we e some misunde s andings a he end o he solu ion.
Figu e 3. Th ee examples o pupils’ ca oons (Models A, B, and C om Figu e 1). The speech bubbles
ha e been ansla ed om Finnish o English.
All he pupils in he s udy succeeded in d awing logical ca oons om hei wo d p oblem. All
h ee pupils c ea ed a eliable con ex in hei wo d p oblem (Table 1) and in hei ca oons (Figu e 3) o
he ma hema ical exp ession. The di ision in o pa s and he sub ac ion ha e acqui ed eal meanings
in he pupil’s own ca oons om hei poin o iews.
4.2. How Did he Pupils Expe ience he Use o W i ing and D awing in Ma hema ics Lea ning?
A e he eaching p ocess, we asked he pupils how hey had expe ienced he mul imodal
app oach in ma hema ics lea ning by a ques ionnai e. The esul s a e shown in Tables 3–5. Mos o
he pupils liked w i ing he wo d p oblem and cons uc ing a la ge s o y abou i . An in e es ing
de ail in Table 3 is ha almos all o he gi ls liked w i ing he mos , whe eas mos o he boys liked
d awing ca oons. A mind map was needed in he planning o he s o y; ha pa o he s udy is
omi ed om his a icle. Mind map making de elops analy ical hinking.
B
“24/6 = 4, 4 – 3 = 1. The cookies we e di ided and e e yone go ou
(cookies). The sis e akes om e e yone else h ee cookies. E e y gi l go
only one cookie.” (Pupil 2).
Educ. Sci. 2017, 7, 9 6 o 9
B
“24/6 = 4, 4 – 3 = 1. The cookies we e di ided and e e yone go ou
(cookies). The sis e akes om e e yone else h ee cookies. E e y gi l
go only one cookie.” (Pupil 2).
C
“24/6 − 3 = 1. Samppa di ides he ice hockey s ics in o six g oups. He
le h ee g oups in he s o age and ook one g oup including ou ice
hockey s icks o he aining camp.” (Two sen ences om Pupil 3 and
he es o he solu ion om o he s).
A he end, pupils d ew ca oons in which hey old he whole s o y: he wo d p oblem and i s
solu ion (Figu e 3). Pupil 1 d ew he ca oon abou how Kaisa di ided 24 apples, Pupil 2 d ew he
ca oon abou how Emma di ided 24 cookies (Case B in Figu e 1), and Pupil 3 d ew he ca oon abou
how Samppa di ided 24 ice hockey s icks (Case C in Figu e 1). The las ca oon, C, is no comple e,
because he e we e some misunde s andings a he end o he solu ion.
Figu e 3. Th ee examples o pupils’ ca oons (Models A, B, and C om Figu e 1). The speech bubbles
ha e been ansla ed om Finnish o English.
All he pupils in he s udy succeeded in d awing logical ca oons om hei wo d p oblem. All
h ee pupils c ea ed a eliable con ex in hei wo d p oblem (Table 1) and in hei ca oons (Figu e 3) o
he ma hema ical exp ession. The di ision in o pa s and he sub ac ion ha e acqui ed eal meanings
in he pupil’s own ca oons om hei poin o iews.
4.2. How Did he Pupils Expe ience he Use o W i ing and D awing in Ma hema ics Lea ning?
A e he eaching p ocess, we asked he pupils how hey had expe ienced he mul imodal
app oach in ma hema ics lea ning by a ques ionnai e. The esul s a e shown in Tables 3–5. Mos o
he pupils liked w i ing he wo d p oblem and cons uc ing a la ge s o y abou i . An in e es ing
de ail in Table 3 is ha almos all o he gi ls liked w i ing he mos , whe eas mos o he boys liked
d awing ca oons. A mind map was needed in he planning o he s o y; ha pa o he s udy is
omi ed om his a icle. Mind map making de elops analy ical hinking.
C
“24/6
−
3 = 1. Samppa di ides he ice hockey s ics in o six g oups. He le
h ee g oups in he s o age and ook one g oup including ou ice hockey
s icks o he aining camp.” (Two sen ences om Pupil 3 and he es o
he solu ion om o he s).
Educ. Sci. 2017, 7, 9 6 o 9
B
“24/6 = 4, 4 – 3 = 1. The cookies we e di ided and e e yone go ou
(cookies). The sis e akes om e e yone else h ee cookies. E e y gi l
go only one cookie.” (Pupil 2).
C
“24/6 − 3 = 1. Samppa di ides he ice hockey s ics in o six g oups. He
le h ee g oups in he s o age and ook one g oup including ou ice
hockey s icks o he aining camp.” (Two sen ences om Pupil 3 and
he es o he solu ion om o he s).
A he end, pupils d ew ca oons in which hey old he whole s o y: he wo d p oblem and i s
solu ion (Figu e 3). Pupil 1 d ew he ca oon abou how Kaisa di ided 24 apples, Pupil 2 d ew he
ca oon abou how Emma di ided 24 cookies (Case B in Figu e 1), and Pupil 3 d ew he ca oon abou
how Samppa di ided 24 ice hockey s icks (Case C in Figu e 1). The las ca oon, C, is no comple e,
because he e we e some misunde s andings a he end o he solu ion.
Figu e 3. Th ee examples o pupils’ ca oons (Models A, B, and C om Figu e 1). The speech bubbles
ha e been ansla ed om Finnish o English.
All he pupils in he s udy succeeded in d awing logical ca oons om hei wo d p oblem. All
h ee pupils c ea ed a eliable con ex in hei wo d p oblem (Table 1) and in hei ca oons (Figu e 3) o
he ma hema ical exp ession. The di ision in o pa s and he sub ac ion ha e acqui ed eal meanings
in he pupil’s own ca oons om hei poin o iews.
4.2. How Did he Pupils Expe ience he Use o W i ing and D awing in Ma hema ics Lea ning?
A e he eaching p ocess, we asked he pupils how hey had expe ienced he mul imodal
app oach in ma hema ics lea ning by a ques ionnai e. The esul s a e shown in Tables 3–5. Mos o
he pupils liked w i ing he wo d p oblem and cons uc ing a la ge s o y abou i . An in e es ing
de ail in Table 3 is ha almos all o he gi ls liked w i ing he mos , whe eas mos o he boys liked
d awing ca oons. A mind map was needed in he planning o he s o y; ha pa o he s udy is
omi ed om his a icle. Mind map making de elops analy ical hinking.
A he end, pupils d ew ca oons in which hey old he whole s o y: he wo d p oblem and i s
solu ion (Figu e 3). Pupil 1 d ew he ca oon abou how Kaisa di ided 24 apples, Pupil 2 d ew he
ca oon abou how Emma di ided 24 cookies (Case B in Figu e 1), and Pupil 3 d ew he ca oon abou
how Samppa di ided 24 ice hockey s icks (Case C in Figu e 1). The las ca oon, C, is no comple e,
because he e we e some misunde s andings a he end o he solu ion.
1
Figu e 3.
Th ee examples o pupils’ ca oons (Models A, B, and C om Figu e 1). The speech bubbles
ha e been ansla ed om Finnish o English.
All he pupils in he s udy succeeded in d awing logical ca oons om hei wo d p oblem.
All h ee pupils c ea ed a eliable con ex in hei wo d p oblem (Table 1) and in hei ca oons
(Figu e 3) o he ma hema ical exp ession. The di ision in o pa s and he sub ac ion ha e acqui ed
eal meanings in he pupil’s own ca oons om hei poin o iews.
Educ. Sci. 2017,7, 9 7 o 9
4.2. How Did he Pupils Expe ience he Use o W i ing and D awing in Ma hema ics Lea ning?
A e he eaching p ocess, we asked he pupils how hey had expe ienced he mul imodal
app oach in ma hema ics lea ning by a ques ionnai e. The esul s a e shown in Tables 3–5. Mos o he
pupils liked w i ing he wo d p oblem and cons uc ing a la ge s o y abou i . An in e es ing de ail in
Table 3is ha almos all o he gi ls liked w i ing he mos , whe eas mos o he boys liked d awing
ca oons. A mind map was needed in he planning o he s o y; ha pa o he s udy is omi ed om
his a icle. Mind map making de elops analy ical hinking.
Table 3. The mos ag eeable pa in he p ojec om he pupils’ poin s o iew (N= 21).
Gi ls Boys
D awing ca oons 1 6
W i ing wo d p oblem and s o y 11 1
Making mind map o a s o y 0 2
The pupils we e asked whe he hey hough ha w i ing ( he use o na u al language) suppo ed
he solu ion making o wo d p oblems. Almos all o he pupils hough ha w i ing suppo ed i
(Table 4). The esul is in e es ing, because w i ing was no a o ed by he boys.
Table 4.
Do you hink ha w i ing (na u al language) suppo s he solu ions o wo d p oblems?
(N= 21).
Answe Gi ls Boys
Yes 12 6
No 0 2
No answe 0 1
The pupils also p o ided hei opinions as o why w i ing is impo an o no in he sol ing
p ocess. By con en analysis, ou main hemes we e ound, which a e p esen ed in Table 5wi h
ex examples.
Table 5.
Examples o pupils
'
opinions on he impo ance o using na u al language in solu ions o
wo d p oblems.
Theme Examples
Be e unde s anding “I unde s and be e how I ha e sol ed he p oblem.” (5 pupils)
Easie o cons uc he solu ion “I is easie o sol e he p oblem when you can w i e.” (5 pupils)
Checking o he answe
“You can jus i y om he w i en ex ha you ha e sol ed i co ec ly” (2 pupils)
Mo he ongue lea ning (w i ing) “You lea n a same ime ma hema ics and mo he ongue” (1 pupil)
5. Discussion
The pedagogical model desc ibed in his pape is an example o using mul imodal ways o
meaning-making in school ma hema ics. The in en ional use o mul imodal languaging model e ealed
he meanings pupils made o he ma hema ical symbolic language and concep s (he e, di ision and
sub ac ion) and se ed in in e p e ing hei hinking. These a ious ypes o ways o exp ess one’s
hinking se e also as a way o he eache o e alua e he s uden s’ unde s anding o he concep s.
Despi e he un amilia i y o he ask, e e y pupil in he class managed o p oduce a p ope wo d
p oblem, solu ions by na u al language, and a pic o ial language o he p oblem and he ca oons.
The mos ypical in e p e a ion o di ision was di ision in o pa s (Figu e 1). F om he poin o iew o
he pupils, w i ing was seen as use ul o a be e unde s anding o he solu ion, and o cons uc ing
he solu ion mo e easily (Tables 4and 5). A simila esul has also been obse ed in he languaging
Educ. Sci. 2017,7, 9 8 o 9
su eys o uni e si y ma hema ics eaching. The gi ls liked w i ing mo e han he boys, and he boys
liked d awing mo e han he gi ls (Table 3). Boys ha ing p oblems in w i ing a e seen on a wide scale
in Finland; na ional Finnish assessmen s o lea ning ou comes a he end o comp ehensi e school
ha e shown ha boys a e signi ican ly wo se han gi ls in w i ing [
24
]. The e we e no di e ences in
he da a be ween how pupils made solu ions ia na u al o pic o ial languages: all pupils could do
bo h o hem mainly co ec ly. All pupils we e able o d aw ca oons, p esen ing bo h he p oblem and
i s solu ion.
We sugges ha he b oadened ways o exp essing ma hema ical hinking may help hose who
s uggle wi h ma hema ics and o whom ma hema ical symbolic language as such is di icul o
comp ehend. The use o w i ing and d awing in p oblem sol ing may also s eng hen he lea ning o
ma hema ical concep s, as he use o di e en modes leads o o ganizing one’s ma hema ical hinking.
In his s udy, all o he pupils we e monolingual Finnish speake s; howe e , he use o mul imodali y
may also suppo , o example, L2 lea ne s o whom he pic o ial language may se e as a way o
unde s and ma hema ical concep s.
The use o di e en modes in lea ning ma hema ics could easily be ex ended o he use o , o
example, ideos. The use o a ious modes and e en digi al echnology would connec he symbolic
ma hema ical language o mo e amilia ways o meaning-making o young s uden s. In a b oade
sense, hese kinds o educa ional applica ions a e closely ela ed o a mul ili e acies app oach and
se e as a way o unde s and he a ious ways o making meaning in odays ex ual en i onmen .
A ew limi a ions o his s udy a e o be men ioned. Fi s o all, he da a was collec ed in a speci ic
educa ional con ex (monolingual Finnish class oom); secondly, he sample size is small, as is common
in case s udies. Ne e heless, he da a showed po en ial in de eloping he mul imodal languaging
model, as i se ed o exp ess hinking.
The mul ili e acies amewo k wi h mul imodali y b ings new insigh s in o school ma hema ics.
Fu he esea ch is needed in o how he in en ional use o di e en modes se es in lea ning speci ic
concep s, such as mul iplica ion, o in mul ilingual class ooms.
Acknowledgmen s:
We would like o hank class eache Ma ja Tuomi o co-ope a ion and o p o iding he
class o his s udy.
Au ho Con ibu ions:
Bo h au ho s con ibu ed o he de elopmen o his pape . Jo ma Jou senlah i and
Pi jo Kulju concie ed and designed he da a collec ion. Jo ma Jou senlah i analysed he da a ega ding he concep
o di ision. Bo h au ho s pa icipa ed o he w i ing o his pape .
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Re e ences
1.
Baue s eld, H. Language games in he ma hema ics class oom: Thei unc ion and hei e ec s. In The
Eme gence o Ma hema ical Meaning: In e ac ion in Class oom Cul u es; Cobb, P., Baue s eld, H., Eds.; E lbaum:
Hillsdale, NJ, USA, 1995; pp. 271–294.
2.
Jou senlah i, J. Kielen äminen ma ema iikan opiskelussa. In Ope aja, Asian un ijuus ja Yh eiskun a.
Ainedidak inen Symposium 7.2.2003; Tu un yliopis on kas a us ie eiden iedekunnan julkaisuja B: 72; Vi a, A.,
Ma ila, O., Eds.; Tu un Ope ajankoulu uslai os: Tu ku, Finland, 2003; pp. 188–196. (In Finnish)
3.
Lemke, J. Ma hema ics in he Middle: Measu e, Pic u e, Ges u e, Sign, and Wo d. In Educa ional Pe spec i es
on Ma hema ics as Semiosis: F om Thinking o In e p e ing o Knowing; Ande son, M., Saenz-Ludlow, A.,
Zellwege , S., Ci a elli, V., Eds.; Legas Publishing: O awa, ON, Canada, 2002; pp. 215–234. (In Finnish)
4. Kalan zis, M.; Cope, B. Li e acies; Camb idge Uni e si y P ess: Camb idge, UK, 2012.
5.
Jou senlah i, J.; Kulju, P. Kieli eo ee inen lähes ymis apa kouluma ema iikan sanallisiin eh ä iin
ja niiden kielenne yihin a kaisuihin. In Toisensa Koh aa a Ainedidak iika ; Tampe een yliopis on
ope ajankoulu uslai oksen julkaisuja A31; Ropo, E., Sil e be g, H., Soini, T., Eds.; Tampe een Yliopis o:
Tampe e, Finland, 2010; pp. 66–77. (In Finnish)
6.
Solano-Flo e, G. Func ion and o m in esea ch on language and ma hema ics educa ion. In Language and
Ma hema ics Educa ion; Moschko ich, J.N., Ed.; IAP-In o ma ion Age: Cha lo e, NC, USA, 2010; pp. 113–149.
Educ. Sci. 2017,7, 9 9 o 9
7.
Jou senlah i, J. Lukiolaisen Teh ä äo ien oi uneen Ma emaa isen Aja elun Pii ei ä: 1990-lu un Pi kän
Ma ema iikan Opiskelijoiden Ma emaa isen Osaamisen ja Uskomus en Ilmen ämänä; Ac a Uni e si a is
Tampe ensis 1061; Tampe een Yliopis o: Tampe e, Finland, 2005. (In Finnish)
8.
Me sämuu onen, J. (Ed.) Pe usope uksen Ma ema iikan Oppimis ulos en Pi ki äisa ioin i Vuosina 2005–2012;
Koulu uksen seu an a apo i 2013: 4; Edi a P ima Oy: Helsinki, Finland, 2013. (In Finnish)
9.
Jou senlah i, J.; Kulju, P.; Tuomi, M. Ma emaa isen lausekkeen kon eks ualisoin i sanalliseksi eh ä äksi
ja a inaksi. Ope uskokeilu ki joi amisen hyödyn ämises ä ma ema iikan opiskelussa. In Ainedidak inen
Tu kimus Koulu uspolii isen Pää öksen eon Pe us ana; Ainedidak isia u kimuksia 4; Tainio, L., Juu i, K.,
Rou a inne, S., Eds.; Suomen Ainedidak inen Tu kimusseu a y: Helsinki, Finland, 2012; pp. 107–122.
(In Finnish)
10.
New London G oup. A pedagogy o mul ili e acies: Designing social u u es. Ha . Educ. Re .
1996
,66,
60–92.
11. K ess, G. Li e acy in he New Media Age; Rou ledge: London, UK, 2003.
12.
Kulju, P.; Kupiainen, R.; Wiseman, A.; Jy kiäinen, A.; Koskinen-Sinisalo, K.-L.; Mäkinen, M. A Re iew o
Mul ili e acies in P ima y Class ooms. 2016, submi ed o publica ion.
13. O’Hallo an, K. Ma hema ical Discou se: Language, Symbolism and Visual Images; Con inuum: London, UK, 2005.
14.
Takeuchi, M. The Si ua ed Mul ili e acies App oach o Class oom Pa icipa ion: English Language Lea ne s’
Pa icipa ion in Class oom Ma hema ics P ac ices. J. Lang. Iden i y Educ. 2015,14, 159–178. [C ossRe ]
15.
Ope ushalli us. Finnish Co e Cu iculum o Basic Educa ion 2014. A ailable online: h p://www.oph. i/
ops2016/pe us ee (accessed on 31 Oc obe 2016).
16.
Palsa, L.; Ruokamo, H. Behind he concep s o mul ili e acies and media li e acy in he enewed Finnish
co e cu iculum: A sys ema ic li e a u e e iew o pee - e iewed esea ch. Semina . ne -In . J. Media Technol.
Li elong Lea n. 2015,11, 101–119.
17.
Mo gan, C. The place o pupil w i ing in lea ning, eaching and assessing ma hema ics. In Issues in
Ma hema ics Teaching; Ga es, P., Ed.; Rou ledge: London, UK, 2001; pp. 232–244.
18.
Ch onaki, A.; Ch is iansen, I. Challenging pe spec i es on ma hema ics class oom communica ion: om
ep esen a ions o con ex , in e ac ions, and poli ics. In Challenging Pe spec i es on Ma hema ics Class oom
Communica ion; Ch onaki, A., Ch is iansen, I., Eds.; IAP-In o ma ion Age: G eenwich, CO, USA, 2005;
pp. 3–48.
19.
Jou senlah i, J.; Ali-Löy y, S.; Pohjolainen, S. De eloping Lea ning and Teaching in Enginee ing
Ma hema ics wi h and wi hou Technology. A ailable online: h p://www.se i.be/con e ence-2016/
pape s/Ma hema ics_and_Enginee ing_Educa ion/jou senlah i-de eloping-lea ning-and- eaching-in-
enginee ing-ma hema ics-wi h-and-wi hou - echnology-153_a.pd (accessed on 31 Oc obe 2016).
20.
Jou senlah i, J.; Kulju, P. Kielen äminen ma ema iikan ja äidinkielen ope uksen kehi ämisessä.
In Moniluku ai o Kaikki Kaikessa; Kaa inen, T., Ed.; Tampe een Yliopis on No maalikoulu: Tampe e, Finland,
2015; pp. 57–76. (In Finnish)
21.
Jou senlah i, J.; Rä yä, K. Kielen ämisen käsi e ainedidak isissa u kimuksissa. In Raja on Tule aisuus: Koh i
Kokonais al ais a Oppimis a: Ainedidak iikan Symposium Jy äskylässä 13-14.2.2014; Suomen ainedidak isen
u kimusseu an julkaisuja 8; Kauppinen, M., Rau iainen, M., Ta nanen, M., Eds.; Suomen Ainedidak inen
Tu kimusseu a y: Helsinki, Finland, 2015; pp. 45–62. (In Finnish)
22.
Simon, M.A. P ospec i e elemen a y eache s’ knowledge o di ision. J. Res. Ma h. Educ.
1993
,24, 233–254.
[C ossRe ]
23.
Fischbein, E.; De i, M.; Nello, M.; Ma ino, M. The ole o implisi models in sol ing e bal p oblems in
mul iplica ion and di ision. J. Res. Ma h. Educ. 1985,16, 3–17. [C ossRe ]
24.
Ha junen, E.; Rau pu o, J. Kielenkäy ön Aja elua ja Aja elun Kielen ämis ä. Äidinkielen ja Ki jallisuuden
Oppimis ulokse Pe usope uksen Pää ö aiheessa 2014: Keskiössä Kielen un emus ja Ki joi aminen; Kansallinen
Koulu uksen A ioin ikeskus: Helsinki, Finland, 2015. (In Finnish)
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