Ci a ion: Ayala-Al ami ano, C.;
Pin o, E.; Molina, M.; Cañadas, M.C.
In e ac ing wi h Inde e mina e
Quan i ies h ough A i hme ic Wo d
P oblems: Tasks o P omo e
Algeb aic Thinking a Elemen a y
School. Ma hema ics 2022,10, 2229.
h ps://doi.o g/10.3390/
ma h10132229
Academic Edi o : Jay Jahangi i
Recei ed: 29 Ap il 2022
Accep ed: 23 June 2022
Published: 25 June 2022
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ma hema ics
A icle
In e ac ing wi h Inde e mina e Quan i ies h ough A i hme ic
Wo d P oblems: Tasks o P omo e Algeb aic Thinking a
Elemen a y School
C is ina Ayala-Al ami ano 1, Ede Pin o 2,* , Ma a Molina 3and Ma ía C. Cañadas 4
1Depa men o Educa ion, Uni e si y o Alme ia, Ca e e a San U bano, La Cañada, 04120 Alme ía, Spain;
[email p o ec ed]
2Facul y o Educa ion, Uni e sidad del Desa ollo, A . Plaza 680, San iago 7610658, Chile
3Depa amen o de Didác ica de las Ma emá icas y de las Ciencias Expe imen ales, Escuela de Educación y
Tu ismo de Á ila, Uni e sidad de Salamanca, Mad igal de las Al as To es, 3, 05003 Á ila, Spain;
[email p o ec ed]
4Depa men o Ma hema ics Didac ics, Facul y o Educa ion Sciences, Uni e si y o G anada,
18071 G anada, Spain; mconsu@ug .es
*Co espondence: [email p o ec ed]
Abs ac :
In his s udy, we analyze how 9–10-yea -old pupils wo k wi h equa ions, a cen al aspec o
algeb aic hinking in ea ly g ades and a co ne s one o mo e o mal lea ning o algeb a. Speci ically,
we seek: (a) o desc ibe he main cha ac e is ics o he asks ha suppo algeb aic hinking h ough a
ansla ion p ocess om a i hme ic wo d p oblems o algeb aic language and ice e sa, and (b) o
iden i y how pupils e e o inde e mina e quan i ies in hese con ex s and wha meaning hey gi e
o hem. The analysis ocuses on he seman ic cong uence o he exp essions p oposed by hem and
on he dialogue hey held du ing he ansla ion p ocess. We analyzed he o al discussion in he
pools and he w i en esponses o he p oblem ha pupils posed. The esul s show ha a i hme ic
wo d p oblems allow he inde e mina e o become an objec o hough o pupils, who ep esen
i in mul iple ways and e e o i when p oposing equa ions ha ep esen he s uc u e o each
p oblem. Ano he inding highligh s ha e lec ion on he in e p e a ion o he equa ions suppo s
he iden i ica ion o wo meanings associa ed wi h inde e mina e quan i ies, namely, unknown
and a iable.
Keywo ds:
a i hme ic wo d p oblems; ea ly algeb a; inde e mina e quan i ies; elemen a y educa ion;
p oblem posing; ansla ion
MSC: 97H20
1. In oduc ion
Cu en ly, di e en cu icula guidelines conside algeb aic hinking as a ans e sal
opic om he beginning o schooling [
1
–
4
]. These p oposals ecommend p omo ing in
pupils he iden i ica ion o gene al ma hema ical ela ionships and s uc u es based on
si ua ions app op ia e o hei age, which a e pa o hei daily expe iences and na u al
in ui ions. Despi e he p esence o algeb aic hinking in he di e en cu icula, he e a e s ill
challenges on how o in oduce his ype o hinking in elemen a y educa ion class ooms.
Speci ically, wi h his s udy, we seek o con ibu e wi h ways o app oaching algeb aic
hinking om con en s ha ha e adi ionally been seen as exclusi ely a i hme ic. In
his pape , we ocus on iden i ying how elemen a y school pupils ep esen and e e
o inde e mina e quan i ies when hey es ablish ela ionships be ween he esolu ion o
a i hme ic wo d p oblems (AWPs, he ea e ) and hei ansla ion using algeb aic language,
and ice e sa. This ansla ion p ocess will allow us o del e in o he pa hs ha pupils
ha e o gi e meaning o he inde e mina e.
Ma hema ics 2022,10, 2229. h ps://doi.o g/10.3390/ma h10132229 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 2229 2 o 18
Inde e mina e quan i ies cons i u e a cen al aspec o algeb aic hinking, which can
be associa ed wi h di e en meanings depending on he con ex . They can be in e p e ed
as a gene alized numbe , an unknown quan i y, a a iable quan i y, o a pa ame e . Ge ing
elemen a y pupils o gene a e ich meanings om he inde e mina e depends on he
lea ning oppo uni ies and di e si y o lea ning expe iences hey a e aced wi h [
5
]. In
his s udy, we add ess how s uden s in e ac wi h he inde e mina e quan i ies h ough
he p ocess o p oblem posing and ansla ing om algeb aic o na u al language, and
ice e sa.
A g owing body o esea ch has shown ha pupils be ween he ages o 6 and 12 e e
o and ep esen inde e mina e quan i ies using mul iple ep esen a ions [
6
,
7
]. Rega d-
ing he use o algeb aic language, and in pa icula he use o le e s, elemen a y pupils
accep i s use and co ec ly ep esen a iable quan i ies by gene alizing ela ionships
be ween quan i ies ha co a y [
5
,
7
–
9
]. I has been e idenced ha eaching and lea ning
en i onmen s ha encou age child en o u ilize non-nume ical symbols o ep esen inde-
e mina e quan i ies, such as a iable no a ion, can help hem cons uc an unde s anding
o a iables [
10
]. Howe e , in he ansi ion o using his ype o no a ion co ec ly, some
e o s and di icul ies e idenced in highe g ades a e eplica ed [
11
–
13
]. Fo example, i
is obse ed ha pupils spon aneously assign alues o li e al symbols acco ding o hei
posi ion in he alphabe , o al hough hey ecognize ha hey can ep esen di e en alues,
hey a ibu e speci ic alues chosen a andom [
13
]. On he o he hand, he li e a u e
ecommends gi ing conc e e meaning o ma hema ical language h ough amilia con ex s
and ecognizing amilia i y as an impo an ac o in he p oblem-sol ing p ocess.
Conside ing he p e ious aspec s, ou p oposal is o ca y ou asks whose objec i e is
o suppo he algeb aic hinking o elemen a y school pupils and he co ec ep esen a ion
o inde e mina e quan i ies conside ing amilia con ex s o hem.
1.1. Algeb aic Thinking
The concep ual amewo k ha di ec s ou s udy conside s ha algeb aic hinking e e s
o inde e mina e quan i ies, and hese quan i ies a e ea ed analy ically, ha is, e en i he
quan i ies a e unknown, hey a e added, sub ac ed, mul iplied, o di ided [
7
]. Mo e speci ically,
algeb aic hinking can be unde s ood as he ou co e p ac ices o gene alizing, ep esen ing,
jus i ying, and easoning wi h ma hema ical s uc u e and ela ionships [14]. Speci ically:
•
Gene alize can be in e p e ed, in a b oad way, as he ac ion o ecognizing ha some
a ibu es o a ma hema ical si ua ion can change, while o he s emain in a iable [
15
].
A ending o gene aliza ion allows pupils o mo e away om he pa icula i ies asso-
cia ed wi h a i hme ic calcula ion and, in u n, allows hem o iden i y he s uc u e
and ma hema ical ela ionships in ol ed in each si ua ion [15].
•
Rep esen ing gene al ma hema ical ideas can in ol e di e en semio ic means, some
con en ional and o he s no , such as ges u es, he hy hm o speaking, and na u al
language [
7
]. The exp ession o gene aliza ion will ha e di e en deg ees o sophis i-
ca ion depending on he means o ep esen a ion used.
•
Jus i ying gene aliza ions equi es pupils o de e mine and explain he u h o a
conjec u e o claim [
7
]. This suppo s a be e unde s anding o he p oblem, i s
s uc u e, and i s ela ionships. P omo ing jus i ica ion in he class oom helps o:
e ine gene aliza ion [
16
]; o pupils o exp ess hemsel es mo e clea ly; and o
eache s o make well-in o med pedagogical decisions since hey can unde s and wha
pupils hink based on wha hey say o he use hey make o signs [17].
•
Reasoning in ol es ea ing gene aliza ions as objec s in hemsel es [
18
], which implies
ha pupils use he gene aliza ions ha hey ha e ound, ep esen ed, and jus i ied in
o he ypes o ma hema ical si ua ions.
The ou co e p ac ices a e embodied in he di e en app oaches o ea ly algeb a:
(a) gene alized a i hme ic, which in ol es gene alizing, ep esen ing, jus i ying, and eason-
ing wi h a i hme ic ela ionships, including undamen al p ope ies o ope a ions as well as
o he ypes o ela ionships on classes o numbe s [
16
]; (b) equi alence,exp essions, equa ions,
Ma hema ics 2022,10, 2229 3 o 18
and inequali ies, which include de eloping a ela ional unde s anding o he equal sign and
gene alizing, ep esen ing, and easoning wi h exp essions, equa ions, and inequali ies,
including in hei symbolic o ms [
14
]; and (c) unc ional hinking, which includes gene aliz-
ing ela ionships be ween co- a ying quan i ies and ep esen ing, jus i ying, and easoning
wi h hese gene aliza ions h ough na u al language, a iable no a ion, d awings, ables,
and g aphs [
18
]. In his s udy, we ocus on he equa ions, i.e., he second con en a ea
desc ibed p e iously.
1.2. Linea Equa ions in Elemen a y School
In his s udy, we ocus on linea equa ions because hese a e deemed sui able o
he age and hei wo k is sugges ed in elemen a y school cu icula [
1
–
4
]. We unde s and
a linea equa ion is a ma hema ical sen ence ha in ol es an equal sign o show ha
wo algeb aic o nume ic exp essions a e equi alen [
14
], wi h one o mo e unknowns.
Rad o d [
19
] poin ed ou ha using an equa ion o eason abou he ep esen a ion and
communica ion o ela ionships be ween quan i ies is a co ne s one o algeb a. In addi ion,
many p oblems a e be e sol ed i he equa ion is i s w i en o ep esen he p oblem
s a emen . He highligh ed ha de eloping an unde s anding o how equa ions can be
w i en o ep esen p oblems a elemen a y school can build a ounda ion o la e lea ning
o o mal algeb a.
1.3. T ansla ion be ween Ve bal Language and Algeb aic Language
AWPs con ain in o ma ion ha is p esen ed exclusi ely h ough na u al language,
and o sol e hem and ind he alue o some unknown quan i y i is necessa y o apply
one o mo e elemen a y ma hema ical ope a ions. Wi hin he amewo k o school algeb a,
AWPs encou age pupils o make sense o he inde e mina e, which does no appea wi hou
suppo as i is a quan i y o some hing ha is no known [
20
]. In his con ex , p oblems
can be ep esen ed using di e en ep esen a ions. Thei in e p e a ion and solu ion can
lead o se e al ansla ions ca ied ou by he sol e .
Rega ding he ansla ion o na u al language o algeb aic language, mos au ho s
ocus mainly on g ades a e elemen a y educa ion. These s udies ha e shown ha o be
success ul in ansla ing be ween na u al language and algeb aic language, elemen a y
pupils mus iden i y he a iables in ol ed, he ela ionships be ween hem, and he syn ax
o he symbolic ep esen a ion. Rega ding he di icul ies ha hey ace, one o hem
is unde s anding he meaning o algeb aic language since his ype o ep esen a ion is
conside ed opaque o hem. They end o ha e di icul y isualizing he ad an ages o
algeb aic language [
11
], so elemen a y school pupils p e e o use a i hme ic- ype s a egies
and ep esen a ions [21].
The e e se ansla ion, om algeb aic language o na u al language, can be conside ed
in he con ex o p oblem posing. This ac i i y equi es pupils o o mula e ma hema ical
p oblems om gi en si ua ions ha may include ma hema ical exp essions o diag ams,
o by e o mula ing exis ing p oblems [
22
]. S oyano a [
23
] p oposes h ee ca ego ies o
p oblem-posing asks: (a) ee si ua ions, (b) semi-s uc u ed si ua ions, and (c) s uc u ed
si ua ions. In his s udy, we ocus on he second ca ego y. These asks a e cha ac e ized
by being based on an open si ua ion, pa icula ly an equa ion. F om his poin , we in i e
elemen a y school pupils o c ea e a p oblem by applying ma hema ical p ocedu es, con-
cep s, and ela ionships om hei own expe iences. This ype o ask is associa ed wi h
high cogni i e demand; whoe e in en s he p oblems mus e lec on he s uc u e o
he si ua ion a he han on he p ocedu es o sol ing he p oblem [
24
]. P e ious s udies
ha e shown ha posing p oblems om gi en ma hema ical equa ions o calcula ions e-
qui es unde s anding he meaning o he ope a ions [
25
]. In addi ion, in his ype o ask,
pupils usually ollow an algo i hmic p ocess ocused on he ope a ional and non-seman ic
s uc u e o he p oblems [26].
Ma hema ics 2022,10, 2229 4 o 18
P oblem posing, om he eaching pe spec i e, is a means o e alua ing he concep-
ions o pupils ega ding a pa icula opic [
22
,
27
], and allows hei abili ies o use hei
ma hema ical knowledge o be ecognized.
2. Resea ch Objec i es
In his s udy, we analyze he wo k comple ed by a g oup o 9–10-yea -old pupils.
Speci ically, we seek o: (a) desc ibe he main cha ac e is ics o he asks ha suppo
algeb aic hinking h ough a ansla ion p ocess om AWP o algeb aic language and ice
e sa, and (b) iden i y how pupils e e o inde e mina e quan i ies in hese con ex s and
wha meaning hey gi e o hem.
3. Ma e ials and Me hods
This s udy is pa o a b oade Class oom Teaching Expe imen (CTE) [
28
], and is pa
o he esea ch-design pa adigm [
29
]. The gene al objec i e o he CTE was o guide pupils
ages 9–10 in he exp ession and jus i ica ion o gene al ma hema ical ideas by wo king on
h ee app oaches o school algeb a.
3.1. Con ex
This s udy was conduc ed in he con ex o a summe school o pupils who had
jus inished 4 h g ade o he elemen a y school. The summe school is an ac i i y o ga-
nized e e y yea by he Facul y o Educa ion o he Uni e sidad del Desa ollo (San iago,
Chile), wi h he aim o p o iding e ec i e and ai oppo uni ies o child en h ough he
de elopmen o hinking and inno a ion.
Bo h he design and he implemen a ion o his ac i i y con empla ed a collabo a i e
wo k ha in ol ed he pa icipa ion o esea che s and eache s om he a ea o ma he-
ma ics educa ion in Chile and Spain. Due o he heal h si ua ion caused by he COVID-19
pandemic, he summe school was de eloped i ually. Speci ically, he pupils accessed
he ac i i ies om hei homes, h ough di e en de ices connec ed o he in e ne : mobile
phones, able s, o compu e s. Addi ionally, each pupil used physical ma e ials ha we e
p o ided by he uni e si y: a boa d wi h ma ke s o di e en colo s, a olde o eco d hei
indings on wo kshee s, and manipula i es. The s uden s we e encou aged o keep he
came as on, so ha he eache – esea che s could obse e hei wo k. In u n, he s uden s’
amilies we e asked o allow hem o wo k on hei own wi hou he help o o he s.
3.2. Pa icipan s
This quali a i e s udy in ol ed 21 pupils who we e be ween 9 and 10 yea s old and
who had comple ed 4 h g ade online, gi en he heal h con ex . The pupils belonged o wo
schools ha a e pa o he same Educa ional Founda ion ha se es child en and young
people om low-income sec o s. Speci ically, nine pupils om one school and 12 om
ano he . Pupils’ anonymi y in his pape was ensu ed by assigning each a code: Si, whe e
i = 1 . . . 21.
The pupils we e selec ed wi h he help o hei egula ma h eache s unde he
ollowing h ee c i e ia: (a) gende pa i y (10 gi ls and 11 boys); (b) willingness o wo k
du ing he summe ; and (c) di e en paces o lea ning.
Rega ding he p e ious knowledge o he pupils, i is impo an o poin ou ha ,
al hough he Chilean cu iculum con empla es a hema ic opic o algeb a and pa e ns [
1
],
he lea ning objec i es ela ed o his opic co esponding o 4 h g ade we e no ully
add essed due o he coun y’s heal h si ua ion [
30
]. In he ma hema ics classes, he
lea ning objec i es ha e e o he opic o numbe s and ope a ions we e mainly add essed.
Rega ding he opic o pa e ns and algeb a, we wo ked wi h nume ical pa e ns ha
in ol e an ope a ion, which was egis e ed in ables. In he p e ious g ade (3 d g ade),
pupils sol ed one-s ep equa ions in ol ing addi ion and sub ac ion and a geome ic
symbol ep esen ing an unknown numbe . They employed s a egies such as ial and
e o o he in e se ope a ion. The ma hema ical ep esen a ions ha hey used we e
Ma hema ics 2022,10, 2229 5 o 18
nume ical–symbolic ones and he emphasis o he classes was ocused on p omo ing
luency in he calcula ion.
3.3. Design
The summe school was o ganized in 10 sessions, including a p e- es and a pos -
es , and ollowing he app oaches o algeb aic hinking [
14
]: (a) gene alized a i hme ic;
(b) equi alence, exp essions, and equa ions; and (c) unc ional hinking (see Figu e 1).
Ma hema ics 2022, 10, x FOR PEER REVIEW 5 o 19
[1], he lea ning objec i es ela ed o his opic co esponding o 4 h g ade we e no ully
add essed due o he coun y’s heal h si ua ion [30]. In he ma hema ics classes, he
lea ning objec i es ha e e o he opic o numbe s and ope a ions we e mainly
add essed. Rega ding he opic o pa e ns and algeb a, we wo ked wi h nume ical
pa e ns ha in ol e an ope a ion, which was egis e ed in ables. In he p e ious g ade
(3 d g ade), pupils sol ed one-s ep equa ions in ol ing addi ion and sub ac ion and a
geome ic symbol ep esen ing an unknown numbe . They employed s a egies such as
ial and e o o he in e se ope a ion. The ma hema ical ep esen a ions ha hey used
we e nume ical–symbolic ones and he emphasis o he classes was ocused on p omo ing
luency in he calcula ion.
3.3. Design
The summe school was o ganized in 10 sessions, including a p e- es and a pos - es ,
and ollowing he app oaches o algeb aic hinking [14]: (a) gene alized a i hme ic; (b)
equi alence, exp essions, and equa ions; and (c) unc ional hinking (see Figu e 1).
Figu e 1. O ganiza ion o sessions.
In he i s and las sessions, he pupils’ esponses o di e en algeb aic asks we e
assessed. Sessions 2–9 ollowed a simila s uc u e, o ganized in o h ee pa s:
• small g oups (4–5 pupils), in which he aim was o he pupils o dialogue and
collabo a e wi h each o he in he sea ch o egula i ies, conjec u es, and solu ions
o he p oblems p esen ed;
• whole g oup, whe e each g oup p esen ed hei indings and wo eache s led he
discussion so ha he pupils syn hesized hei ideas; and
• medium-sized g oups (10–11 pupils), in which he objec i e was o ans e wha had
been discussed o ano he simila si ua ion o o del e in o a inding om he
p e ious pa s on he p oblems p esen ed. Each pa suppo ed he ins alla ion o
spaces o coope a ion, con on a ion, and discussion o ideas.
3.4. Ins uc ion Sequence: Sessions 2–5
In his s udy, we ocus on session 6. Howe e , we also desc ibe, in gene al e ms,
wha happened p e iously, wi hou conside ing he ini ial assessmen . In session 2, pupils
exp essed hei gene al ideas h ough na u al language by a guing wha happens when
odd and e en numbe s a e added. Then, in sessions 3 and 4, hey discussed he meaning
o he equal sign. He e, o he i s ime, he le e is in oduced as a ep esen a ion o
gene alizing a i hme ic p ope ies ( o example, he commu a i i y o addi ion). In his
i s encoun e wi h le e s, he pupils concluded ha i could ep esen any numbe
(gene alized numbe ). In he i h and six h sessions, we ocused on he exp ession and
esolu ion o equa ions. Du ing hese sessions we sugges ed o he pupils o (a) ep esen
and sol e equa ions using di e en s a egies, (b) use he le e as a ep esen a ion o an
unknown, and (c) p o ide e idence o alida e gi en explana ions. We in oduced he
ideas o equali y, equa ion, ma hema ical his o ies (an idea we use o e e o he
Figu e 1. O ganiza ion o sessions.
In he i s and las sessions, he pupils’ esponses o di e en algeb aic asks we e
assessed. Sessions 2–9 ollowed a simila s uc u e, o ganized in o h ee pa s:
•
small g oups (4–5 pupils), in which he aim was o he pupils o dialogue and collab-
o a e wi h each o he in he sea ch o egula i ies, conjec u es, and solu ions o he
p oblems p esen ed;
•
whole g oup, whe e each g oup p esen ed hei indings and wo eache s led he
discussion so ha he pupils syn hesized hei ideas; and
•
medium-sized g oups (10–11 pupils), in which he objec i e was o ans e wha had
been discussed o ano he simila si ua ion o o del e in o a inding om he p e ious
pa s on he p oblems p esen ed. Each pa suppo ed he ins alla ion o spaces o
coope a ion, con on a ion, and discussion o ideas.
3.4. Ins uc ion Sequence: Sessions 2–5
In his s udy, we ocus on session 6. Howe e , we also desc ibe, in gene al e ms,
wha happened p e iously, wi hou conside ing he ini ial assessmen . In session 2, pupils
exp essed hei gene al ideas h ough na u al language by a guing wha happens when
odd and e en numbe s a e added. Then, in sessions 3 and 4, hey discussed he meaning
o he equal sign. He e, o he i s ime, he le e is in oduced as a ep esen a ion
o gene alizing a i hme ic p ope ies ( o example, he commu a i i y o addi ion). In
his i s encoun e wi h le e s, he pupils concluded ha i could ep esen any numbe
(gene alized numbe ). In he i h and six h sessions, we ocused on he exp ession and
esolu ion o equa ions. Du ing hese sessions we sugges ed o he pupils o (a) ep esen
and sol e equa ions using di e en s a egies, (b) use he le e as a ep esen a ion o an
unknown, and (c) p o ide e idence o alida e gi en explana ions. We in oduced he
ideas o equali y, equa ion, ma hema ical his o ies (an idea we use o e e o he ansla ion
om na u al language o algeb aic language), and le e s as unknowns, among o he s.
In session 5, hey p oposed an adap a ion o he ca ds and en elopes p oblem desc ibed
in [
20
] (Figu e 2). The pupils we e asked o exp ess and sol e he equa ion in ol ed in
he p oblem using manipula i es and pic o ial ep esen a ions. Howe e , i became clea
ha i was necessa y o deepen he app oach o equa ions. Al hough hey we e able o
ep esen hem wi h manipula i e ma e ial, d awings, and e en le e s, doub s emained
abou he meaning o he inde e mina e.
Ma hema ics 2022,10, 2229 6 o 18
Ma hema ics 2022, 10, x FOR PEER REVIEW 6 o 19
ansla ion om na u al language o algeb aic language), and le e s as unknowns, among
o he s. In session 5, hey p oposed an adap a ion o he ca ds and en elopes p oblem
desc ibed in [20] (Figu e 2). The pupils we e asked o exp ess and sol e he equa ion
in ol ed in he p oblem using manipula i es and pic o ial ep esen a ions. Howe e , i
became clea ha i was necessa y o deepen he app oach o equa ions. Al hough hey
we e able o ep esen hem wi h manipula i e ma e ial, d awings, and e en le e s,
doub s emained abou he meaning o he inde e mina e.
Figu e 2. P oblem session 5.
3.5. Session 6
In he i s pa o he session, wo p oblems we e p esen ed o he pupils:
• I bough a box o colo ed pencils. A home I had pencils, now I ha e 20 in o al. How
many pencils a e in he box? and
• I ha e a baske o apples. Inside he baske , he e a e 20 g een apples and o he ed
ones. How many apples a e in he baske ?
These AWPs we e ep esen ed wi h na u al language and he pupils had o ansla e
and ep esen wi h algeb aic language. The i s AWP in ol es an unknown, has a unique
solu ion, and in ol es he s uc u e y + 15 = 20. The o he AWP implica es wo unknowns
whose alues could no be de e mined due o he lack o da a in he s a emen , and i
in ol es he s uc u e y = 20 + b.
The pupils had o ep esen he AWPs on hei boa ds. Fi s ly, he pupils we e asked
o “ ell he s o y” (i.e., o ep esen e bal sen ences) wi h ma hema ical symbols. We we e
in e es ed in pupils ep esen ing inde e mina e quan i ies howe e hey wan ed. One by
one hey explained how hey did i and discussed whe he wha hei classma es did was
he same as wha each one ep esen ed. Then he possibili y o ep esen ing hem wi h
le e s was men ioned by he eache and discussed wi hin he g oup, as had happened in
p e ious sessions. In Figu e 3 he AWP and ep esen a ions ha he eache aised o
each si ua ion a e p esen ed. The eache p esen ed wo di e en ep esen a ions o each
p oblem: he d awing o he box and colo ed pencils o he baske o apples, and a
symbolic–algeb aic exp ession, wi h he use o le e s. I was belie ed ha amilia
ep esen a ions o he pupils, such as d awings, would allow hem o unde s and he
s a emen o he p oblem and he meaning o each elemen o he algeb aic exp ession.
A e eaching an ag eemen on he ep esen a ion o he p oblems in a g oup discussion,
pupils sol ed each equa ion and discussed wha he alue o each unknown was.
Figu e 3. P oblem session 6.
Figu e 2. P oblem session 5.
3.5. Session 6
In he i s pa o he session, wo p oblems we e p esen ed o he pupils:
•
I bough a box o colo ed pencils. A home I had pencils, now I ha e 20 in o al. How
many pencils a e in he box? and
•
I ha e a baske o apples. Inside he baske , he e a e 20 g een apples and o he ed
ones. How many apples a e in he baske ?
These AWPs we e ep esen ed wi h na u al language and he pupils had o ansla e
and ep esen wi h algeb aic language. The i s AWP in ol es an unknown, has a unique
solu ion, and in ol es he s uc u e y+ 15 = 20. The o he AWP implica es wo unknowns
whose alues could no be de e mined due o he lack o da a in he s a emen , and i
in ol es he s uc u e y= 20 + b.
The pupils had o ep esen he AWPs on hei boa ds. Fi s ly, he pupils we e asked
o “ ell he s o y” (i.e., o ep esen e bal sen ences) wi h ma hema ical symbols. We we e
in e es ed in pupils ep esen ing inde e mina e quan i ies howe e hey wan ed. One by
one hey explained how hey did i and discussed whe he wha hei classma es did was
he same as wha each one ep esen ed. Then he possibili y o ep esen ing hem wi h
le e s was men ioned by he eache and discussed wi hin he g oup, as had happened
in p e ious sessions. In Figu e 3 he AWP and ep esen a ions ha he eache aised o
each si ua ion a e p esen ed. The eache p esen ed wo di e en ep esen a ions o each
p oblem: he d awing o he box and colo ed pencils o he baske o apples, and a symbolic–
algeb aic exp ession, wi h he use o le e s. I was belie ed ha amilia ep esen a ions
o he pupils, such as d awings, would allow hem o unde s and he s a emen o he
p oblem and he meaning o each elemen o he algeb aic exp ession. A e eaching an
ag eemen on he ep esen a ion o he p oblems in a g oup discussion, pupils sol ed each
equa ion and discussed wha he alue o each unknown was.
Ma hema ics 2022, 10, x FOR PEER REVIEW 6 o 19
ansla ion om na u al language o algeb aic language), and le e s as unknowns, among
o he s. In session 5, hey p oposed an adap a ion o he ca ds and en elopes p oblem
desc ibed in [20] (Figu e 2). The pupils we e asked o exp ess and sol e he equa ion
in ol ed in he p oblem using manipula i es and pic o ial ep esen a ions. Howe e , i
became clea ha i was necessa y o deepen he app oach o equa ions. Al hough hey
we e able o ep esen hem wi h manipula i e ma e ial, d awings, and e en le e s,
doub s emained abou he meaning o he inde e mina e.
Figu e 2. P oblem session 5.
3.5. Session 6
In he i s pa o he session, wo p oblems we e p esen ed o he pupils:
• I bough a box o colo ed pencils. A home I had pencils, now I ha e 20 in o al. How
many pencils a e in he box? and
• I ha e a baske o apples. Inside he baske , he e a e 20 g een apples and o he ed
ones. How many apples a e in he baske ?
These AWPs we e ep esen ed wi h na u al language and he pupils had o ansla e
and ep esen wi h algeb aic language. The i s AWP in ol es an unknown, has a unique
solu ion, and in ol es he s uc u e y + 15 = 20. The o he AWP implica es wo unknowns
whose alues could no be de e mined due o he lack o da a in he s a emen , and i
in ol es he s uc u e y = 20 + b.
The pupils had o ep esen he AWPs on hei boa ds. Fi s ly, he pupils we e asked
o “ ell he s o y” (i.e., o ep esen e bal sen ences) wi h ma hema ical symbols. We we e
in e es ed in pupils ep esen ing inde e mina e quan i ies howe e hey wan ed. One by
one hey explained how hey did i and discussed whe he wha hei classma es did was
he same as wha each one ep esen ed. Then he possibili y o ep esen ing hem wi h
le e s was men ioned by he eache and discussed wi hin he g oup, as had happened in
p e ious sessions. In Figu e 3 he AWP and ep esen a ions ha he eache aised o
each si ua ion a e p esen ed. The eache p esen ed wo di e en ep esen a ions o each
p oblem: he d awing o he box and colo ed pencils o he baske o apples, and a
symbolic–algeb aic exp ession, wi h he use o le e s. I was belie ed ha amilia
ep esen a ions o he pupils, such as d awings, would allow hem o unde s and he
s a emen o he p oblem and he meaning o each elemen o he algeb aic exp ession.
A e eaching an ag eemen on he ep esen a ion o he p oblems in a g oup discussion,
pupils sol ed each equa ion and discussed wha he alue o each unknown was.
Figu e 3. P oblem session 6.
Figu e 3. P oblem session 6.
In he second pa o he session, he eache gene a ed a space o discussion wi h he
pupils abou he disco e ies ob ained in he p e ious pa . The discussion sough : (a) o
in es iga e he use o he le e when exp essing he equa ions, and (b) o collec e idence
o de e mine whe he he le e ep esen s an unknown quan i y o a a iable quan i y.
The eache asked ques ions such as: Does he exp ession X+ 15 = 20 ep esen he pencil
p oblem? Does he exp ession T= 20 + S ep esen he apple p oblem? Do you know how
many pencils a e in he box? Can we know how many apples a e in he baske ? How
do you know? Wha e idence do you ha e? Could you explain i in a di e en way o
you pa ne ?
Ma hema ics 2022,10, 2229 7 o 18
A he end o his pa , pupils we e asked o answe a es ia a i ual es o assess
whe he hey could ans e wha was discussed o o he si ua ions wi h simila cha ac-
e is ics. They we e asked o choose he al e na i e ha allowed hem o ell each o he
s o ies shown in Figu e 4. In he second si ua ion, he e we e wo possible co ec answe s:
40 + =jand j+ 40 = , he objec i e was o discuss he easons o why bo h we e co ec .
Ma hema ics 2022, 10, x FOR PEER REVIEW 7 o 19
In he second pa o he session, he eache gene a ed a space o discussion wi h
he pupils abou he disco e ies ob ained in he p e ious pa . The discussion sough : (a)
o in es iga e he use o he le e when exp essing he equa ions, and (b) o collec
e idence o de e mine whe he he le e ep esen s an unknown quan i y o a a iable
quan i y. The eache asked ques ions such as: Does he exp ession X + 15 = 20 ep esen
he pencil p oblem? Does he exp ession T = 20 + S ep esen he apple p oblem? Do you
know how many pencils a e in he box? Can we know how many apples a e in he baske ?
How do you know? Wha e idence do you ha e? Could you explain i in a di e en way
o you pa ne ?
A he end o his pa , pupils we e asked o answe a es ia a i ual es o assess
whe he hey could ans e wha was discussed o o he si ua ions wi h simila
cha ac e is ics. They we e asked o choose he al e na i e ha allowed hem o ell each
o he s o ies shown in Figu e 4. In he second si ua ion, he e we e wo possible co ec
answe s: 40 + = j and j + 40 = , he objec i e was o discuss he easons o why bo h we e
co ec .
Figu e 4. Tes pa 2.
Finally, in he hi d pa , we wan ed he pupils o ex end wha hey had lea ned o
o he cases and hey we e asked o ca y ou he e e se p ocess, ha is, o c ea e
ma hema ical s o ies om he equa ion 25 + u = 45. Each pupil w o e he p oblem ha
each one in en ed on he boa d and hen discussed he ele ance o each si ua ion.
3.6. Analysis
We analyzed pupils’ esponses o (a) desc ibe he cha ac e is ics o he asks ha
suppo algeb aic hinking, and (b) iden i y how hey e e o inde e mina e quan i ies.
The au ho s o his s udy classi ied he esponses conce ning he i s wo pa s o he
session, and all he w i en and o al esponses o he pupils o he AWPs p esen ed and
he in en ed p oblems in he hi d pa . Disc epancies we e hen discussed un il an
ag eemen was eached.
To analyze he pupils’ ansla ions, we based ou me hod on he ideas o Du al’s [31]
p oposal. He poin ed ou ha wo ep esen a ions a e cong uen when he ollowing h ee
condi ions a e me : (a) seman ic co espondence be ween he signi ican uni s ha
cons i u e hem; (b) seman ic uni oci y, i.e., each ini ial signi ican uni o ou pu
co esponds o one and only one signi ican elemen a y uni o he inpu eco d; and (c)
he o de wi hin he o ganiza ion o he signi ican ou pu uni s is main ained in he
a i al ep esen a ion. When one o hese c i e ia is no longe me , he ep esen a ions a e
no cong uen wi h each o he . Howe e , his au ho added ha wo exp essions can be
e e en ially equi alen wi hou being seman ically cong uen . Seman ic cong uence
allows us o see he deg ee o anspa ency o he ela ionship be ween wo
ep esen a ions.
Figu e 4. Tes pa 2.
Finally, in he hi d pa , we wan ed he pupils o ex end wha hey had lea ned
o o he cases and hey we e asked o ca y ou he e e se p ocess, ha is, o c ea e
ma hema ical s o ies om he equa ion 25 + u= 45. Each pupil w o e he p oblem ha each
one in en ed on he boa d and hen discussed he ele ance o each si ua ion.
3.6. Analysis
We analyzed pupils’ esponses o (a) desc ibe he cha ac e is ics o he asks ha
suppo algeb aic hinking, and (b) iden i y how hey e e o inde e mina e quan i ies.
The au ho s o his s udy classi ied he esponses conce ning he i s wo pa s o he
session, and all he w i en and o al esponses o he pupils o he AWPs p esen ed and he
in en ed p oblems in he hi d pa . Disc epancies we e hen discussed un il an ag eemen
was eached.
To analyze he pupils’ ansla ions, we based ou me hod on he ideas o Du al’s [
31
]
p oposal. He poin ed ou ha wo ep esen a ions a e cong uen when he ollowing h ee
condi ions a e me : (a) seman ic co espondence be ween he signi ican uni s ha cons i-
u e hem; (b) seman ic uni oci y, i.e., each ini ial signi ican uni o ou pu co esponds o
one and only one signi ican elemen a y uni o he inpu eco d; and (c) he o de wi hin
he o ganiza ion o he signi ican ou pu uni s is main ained in he a i al ep esen a ion.
When one o hese c i e ia is no longe me , he ep esen a ions a e no cong uen wi h each
o he . Howe e , his au ho added ha wo exp essions can be e e en ially equi alen
wi hou being seman ically cong uen . Seman ic cong uence allows us o see he deg ee o
anspa ency o he ela ionship be ween wo ep esen a ions.
I only co espondence and seman ic uniqueness we e obse ed, he ansla ions we e
classi ied as equi alen .
Seman ically consis en ansla ions and hei equi alen s we e conside ed co ec
ansla ions. Rega ding he inco ec ansla ions, we iden i ied ou ypes o e o s: (a) in-
comple e ansla ion because some da a o he s a emen a e missing; (b) ansla ion ela ed
o he p oblem bu ha does no e e o inde e mina e quan i ies and includes he answe
o he p oblem (e.g., 15 + 5 = 20); (c) appa en ly co ec ansla ion, bu he co espondence
be ween he seman ic uni s poo ly ela es he e ms o he equa ion o he s a emen ; and
(d) ansla ion wi h in en ed in o ma ion o in o ma ion un ela ed o he p oblem.
When ansla ing a gi en s a emen in o na u al language, he pupils’ esponses we e
classi ied in o: (a) pic o ial ansla ion i d awings we e used; (b) symbolic–a i hme ic ans-
la ion i using numbe s and ma hema ical symbols; and (c) algeb aic language ansla ion
Ma hema ics 2022,10, 2229 8 o 18
i i used numbe s, ma hema ical symbols, and some symbols o e e o inde e mina e
quan i ies (le e s o “?”).
To analyze he in en ed p oblems, we ansla ed he p oblems posed in o algeb aic
language, main aining he s uc u e o he gi en equa ion, in addi ion o a le – igh
cong uence whene e possible. Then we compa ed he ansla ion ob ained wi h he
p oposed equa ion. We also iden i ied i he pupils p oposed a new con ex o adap ed he
p oblems p oposed in he p e ious pa s.
To iden i y how he pupils e e ed o inde e mina e quan i ies, we analyzed he o al
discussions in he pools and he w i en eco ds o he p oblems hey in en ed. We looked
o linguis ic exp essions ha used inde ini e adjec i es (e.g., li le, a lo , one, ano he , oo
many, same, some, none, any), key ph ases ha con eyed ha he pupils ecognized an
inde e mina e amoun in he analyzed con ex s. Fo example: “i is a numbe ha we do
no know”, and “i is he numbe you wan ”, among o he s. In p e ious esea ch, such
as [
5
,
15
], elemen a y pupils e e ed o inde e mina e quan i ies wi h he keywo ds “many”
o “in ini e” o he ph ase “ he numbe you wan ”.
On he meaning gi en o inde e mina e quan i ies, we analyzed he answe s gi en
o he ques ions Can we know he answe o he p oblem? Can we know wha quan i y
he le e ep esen s? In hese cases, i he pupils answe ed ha he p oblem only had one
solu ion, we in e p e ed ha he meaning associa ed wi h he inde e mina e quan i y was
unknown. While i hey answe ed ha he answe o he p oblem could no be known
because he e was no enough in o ma ion and ha he answe depended on some da a o
he p oblem, we in e p e ed ha he meaning gi en was ha o a a iable quan i y.
4. Resul s
In his sec ion we i s p esen he answe s o he pupils o he asks in which hey
had o ansla e e bal s a emen s in o algeb aic language. Then, he answe s gi en in he
in e se p ocess, ansla ion om algeb aic language o na u al language, will be discussed.
4.1. F om Na u al Language o Algeb aic Language: The Pencil P oblem
The i s p oblem in oduced in ol ed he s uc u e y+ 15 = 20. In his ins ance,
nine een elemen a y school pupils pa icipa ed in i s esolu ion. Ini ially, he ocus was
on he ansla ion o he s a emen in o algeb aic language. A e his, as he las s ep,
he unknown alue was discussed. Table 1shows a cha ac e iza ion o he ansla ions
p oposed by he pupils. Thi een pupils p oposed a ansla ion using algeb aic language,
one made a symbolic–a i hme ic ansla ion, ou p oposed a pic o ial ansla ion, and one
pupil did no espond.
Table 1. Equa ions p oposed by he pupils in he pencil p oblem (s uc u e y+ 15 = 20).
T ansla ion Pupil Equa ion
S uc u e Equa ion P oposed
by he Pupils
Co ec
T ansla ion
Seman ic Consis ency
SC SU OA
Algeb aic language
S15 and S16 y+ 15 = 20 ? + 15 = 20 Yes Yes Yes Yes
S5a+ 15 = 20 Yes Yes Yes Yes
S3b+ 15 = 20 Yes Yes Yes Yes
S14
15 + y= 20
15 + ? = 20 Yes Yes Yes No
S1and S13 15 + a= 20 Yes Yes Yes No
S7, S8, S17 and S18 15 + x= 20 Yes Yes Yes No
S915 + a= 20 1No No Yes No
S615 + y= 35 15 + a= 35 No No Yes No
Symbolic–a i hme ic S12 15 + 5 = 20 No No Yes No
Pic o ial S2, S4, S10 and S11 No No Yes No
No esponding S19
SC = seman ic co espondence; SU = seman ic uni oci y; OA = o de o app ehension. No e.
1
The equa ion is
inco ec because he pupil says ha he bough 15 pencils and “a” ep esen s he ones he had a home.
Ma hema ics 2022,10, 2229 9 o 18
Ini ially, pupils we e asked o ep esen he p oblem s a emen eely on he boa d.
Ele en o he pupils co ec ly ansla ed he na u al language exp ession in o algeb aic
language. Fou o hem o mula ed an equa ion consis en wi h he pencil p oblem, while
ano he se en p oposed e e en ially equi alen equa ions, bu hese we e no consis en
since hey did no mee he o de o app ehension c i e ion. In his case, he s uc u e
ep esen ed by hem was 15 + y= 20, whose e bal s a emen would ha e o men ion i s
he numbe o pencils hey ha e (15) and hen he numbe o pencils in he box (y).
In he design o he ask, i was conside ed impo an ha he pupils e lec on he
equi alence be ween he exp essions, ha is why, in he discussion, he answe s ha
in ol ed he equa ions in he o ms y+ 15 = 20 and 15 + y= 20 we e con as ed. They
we e asked i hey ep esen ed he same hing, e en hough he o de o he addends was
di e en . In his ega d, S
8
poin ed ou : “i we change hei o de , he esul does no
change due o he commu a i e p ope y”. S
15
also men ioned he commu a i e p ope y
o jus i y he equi alence be ween he exp essions.
Ano he impo an aspec conside ed in he design o he ask was o guide he
discussion so ha he elemen a y pupils explici ly made he co espondence be ween he
e ms o he equa ion and he e bal s a emen . Focusing on inde e mina e quan i ies
and hei ep esen a ion, 10 pupils used le e s (a,b, and x) and ano he ou used he
ques ion ma k “?” sign. In he discussion, he pupils e e ed o he le e s o o he sign
“?” as he numbe o pencils in he box hey do no know. Fo example, S
3
, who p oposed
he equa ion b+ 15 = 20, said: “I hough . He says ha he bough a box, bu you don’
know how many pencils ha box has, and ha ep esen s “b”. 15 a home. I used “b”,
bu any le e can be any numbe .” This las sen ence highligh s ha he accep ed ha he
ep esen a ions o his classma es we e also co ec , e en i di e en le e s we e used.
The e e ence o inde e mina e quan i ies was no only made in he algeb aic language
ansla ions, in his p oblem i was also e idenced in a pic o ial ansla ion (see Figu e 5).
S
2
ep esen ed he inde e mina e quan i y on he d awing using he symbol “?”. She
explained ha she ep esen ed he 15 pencils and a box o mys e y pencils. Howe e ,
he ansla ion is incomple e since she does no e e o o ep esen he o al numbe o
pencils (20). The pic o ial ep esen a ions o o he pupils do no e e o inde e mina e
quan i ies ei he .
Ma hema ics 2022, 10, x FOR PEER REVIEW 10 o 19
Figu e 5. T ansla ion o he e bal s a emen using d awings o S
2
.
The las aspec conside ed in he design was he discussion abou he alue ha he
inde e mina e quan i y could ake. In his p oblem, he pupils ag eed ha he e could only
be i e pencils in he box. They complemen ed hei a gumen by eplacing he le e o
he ques ion ma k “?” wi h ha numbe and sol ing he sum o e i y ha he esul was
20. In his way, hey ea i med hei idea ha he inde e mina e quan i y can ha e only
one alue.
4.2. F om Na u al Language o Algeb aic Language: The Apple P oblem
The second p oblem discussed was ha o apples, whose s uc u e is y = 20 + b. As in
he p e ious p oblem, he ansla ion was ca ied ou i s and hen he possible alue o
each o he inde e mina e quan i ies was discussed. Table 2 shows he cha ac e iza ion o
he pupils’ ansla ions. On his occasion, he numbe o co ec ansla ions wi h algeb aic
language was less han in he p e ious p oblem (9 ou o 19); howe e , he numbe o
pupils who made a ansla ion o his ype was he same. One pupil made a symbolic–
a i hme ic ansla ion, h ee pupils a pic o ial ansla ion, and wo pupils did no answe .
Table 2. Equa ions p oposed by he pupils in he apple p oblem (s uc u e y = 20 + b).
T ansla ion Pupil
Equa ion
S uc u e
Equa ion P oposed by
he Pupils
Co ec
T ansla ion
Seman ic
Consis ency
SC US OA
Algeb aic language
S
5
b + 20 = y a + 20 = ? Yes Yes Yes No
S
13
c + 20 = ? Yes Yes Yes No
S
15
20 + b = y 20 + a = ? Yes Yes Yes No
S
2
, S
6
and S
12
20 + x = ? Yes Yes Yes No
S
3
20 + c = ? Yes Yes Yes No
S
8
20 + b = x Yes Yes Yes No
S
9
20 + x = a Yes Yes Yes No
S
7
and S
17
20 + b = b 20 + x = x No Yes No No
S
4
20 + b = 40 20 + n = 40 No No Yes No
S
18
20 + x = No No Yes No
Symbolic–a i hme ic S
16
10 + 10 = 20 No No No No
Pic o ial S
1
, and S
11
No No Yes No
S
10
No No No No
No esponding S
14
and S
19
SC = seman ic co espondence; SU = seman ic Uni oci y; OA = o de o app ehension.
Figu e 5. T ansla ion o he e bal s a emen using d awings o S2.
The las aspec conside ed in he design was he discussion abou he alue ha he
inde e mina e quan i y could ake. In his p oblem, he pupils ag eed ha he e could only
be i e pencils in he box. They complemen ed hei a gumen by eplacing he le e o
he ques ion ma k “?” wi h ha numbe and sol ing he sum o e i y ha he esul was
20. In his way, hey ea i med hei idea ha he inde e mina e quan i y can ha e only
one alue.
Ma hema ics 2022,10, 2229 16 o 18
no a ion. The exp essions ha we iden i ied in his s udy could be used in o he con ex s o
highligh he inde e mina e quan i y and pupils can ecognize i in di e en con ex s.
In addi ion, he cha ac e is ics o he asks p oposed in his s udy allowed he elemen-
a y pupils o associa e he inde e mina e quan i ies wi h wo di e en meanings. As an
unknown quan i y wi h a ixed alue when hey had enough in o ma ion in he s a emen
abou i , o as a a iable numbe ha depends on he alues gi en o some o he unknown
quan i ies. This dis inc ion is o g ea ele ance gi en he complexi y o he polysemy o
inde e mina e quan i ies in algeb aic con ex s.
As we ha e men ioned in ou concep ual amewo k, in p e ious esea ch wi h ele-
men a y pupils who needed o e e o o gi e meaning o inde e mina e quan i ies
[5,8,9]
,
pupils showed a endency o assign speci ic alues o hem and make mis akes ha a e
also e iden in highe g ades [
11
,
12
]. In his case, i is impo an o highligh ha his did
no occu ; we assume ha i was because he p oposed si ua ions we e associa ed wi h
con ex s ha we e close o he pupils and in ol ed numbe s ha allowed calcula ions
o be ca ied ou easily. The pupils ga e meaning o he inde e mina e conside ing ha
i was some hing ha is no known in his o y, an elemen ha hey also highligh ed in
hei s udy [
18
]. This helped hem ocus on he in o ma ion gi en in he s a emen and
a oid choosing numbe s a andom o eso ing o he alphabe . I is wo h men ioning
ha his ype o e o was obse ed in session 4, in which he le e was p esen ed as a
ep esen a ion o inde e mina e quan i ies o he i s ime and in a ma hema ical con ex
when ep esen ing some p ope ies o he addi ion. One limi a ion o ou s udy is ha i
did no in es iga e how he p e ious di icul ies we e o e come un il eaching he session
ha we discuss in his wo k. An open line o wo k is o analyze he ela ionship be ween
he di e en a eas o algeb a and ca y ou a global analysis o all he sessions, which will
allow he in o ma ion p esen ed he e o be comple ed, desc ibing how he di e en asks
p oposed cause di e en o simila ways o e e ing o inde e mina e quan i ies and gi e
meaning o hem.
Au ho Con ibu ions:
Concep ualiza ion, C.A.-A., E.P., M.M. and M.C.C.; me hodology, C.A.-A.,
E.P., M.M. and M.C.C.; alida ion, C.A.-A., E.P., M.M. and M.C.C.; o mal analysis, C.A.-A., E.P., M.M.
and M.C.C.; in es iga ion, C.A.-A., E.P., M.M. and M.C.C.; esou ces, C.A.-A., E.P., M.M. and M.C.C.;
da a cu a ion, C.A.-A., E.P., M.M. and M.C.C.; w i ing—o iginal d a p epa a ion, C.A.-A., E.P., M.M.
and M.C.C.; w i ing— e iew and edi ing, C.A.-A., E.P., M.M. and M.C.C.; isualiza ion, C.A.-A.,
E.P., M.M. and M.C.C.; supe ision, C.A.-A., E.P., M.M. and M.C.C.; p ojec adminis a ion, E.P. and
M.C.C.; unding acquisi ion, E.P. and M.C.C. All au ho s ha e ead and ag eed o he published
e sion o he manusc ip .
Funding:
This wo k has been de eloped wi hin he p ojec 23.400.173, inanced by Uni e sidad
del Desa ollo (Chile), and he p ojec wi h e e ence PID2020-113601GB-I00, inanced by he S a e
Resea ch Agency (SRA) om Spain.
Ins i u ional Re iew Boa d S a emen :
The s udy was conduc ed in acco dance wi h he Decla a ion
o Helsinki, and app o ed by he E hics Commi ee o Uni e sidad del Desa ollo (CEEII) (da e o
app o al 17 Decembe 2020).
In o med Consen S a emen :
In o med consen was ob ained om all subjec s in ol ed in he s udy.
Da a A ailabili y S a emen : Da a a e a ailable upon eques due o es ic ions.
Con lic s o In e es :
The au ho s decla e no con lic o in e es . The unde s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip ; o
in he decision o publish he esul s.
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