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Spanish p ima y school s uden s’ engagemen wi h a non- adi ional
me hod when adding and sub ac ing: Ri ualised e sus explo a o y
pa icipa ion
Inés Gallego-Sáncheza*, Ve ónica Ma ín-Molinaa, Isabel Ca o-To óa and
José Ma ía Ga ilán-Izquie doa
aDepa amen o de Didác ica de las Ma emá icas, Uni e sidad de Se illa, Se illa, Spain
Co espondence de ails: Depa amen o de Didác ica de las Ma emá icas, Facul ad de
Ciencias de la Educación, Uni e sidad de Se illa, Calle Pi o ecnia, s/n, 41013, Se illa,
Spain.
*Email: [email p o ec ed]
ORCIDs
Inés Gallego-Sánchez h ps://o cid.o g/0000-0002-1088-3172
Ve ónica Ma ín-Molina h p://o cid.o g/0000-0002-6359-5246
Isabel Ca o-To ó h ps://o cid.o g/0000-0002-1539-7333
José Ma ía Ga ilán-Izquie do h ps://o cid.o g/0000-0002-3369-5377
Sho Biog aphical No es o he au ho s:
Inés Gallego-Sánchez is an assis an p o esso o ma hema ics educa ion a he Uni e sidad de
Se illa, Spain. She is a ma hema ician and has a Ph.D. in applied ma hema ics. She is cu en ly
esea ching ma hema ics eaching and lea ning h ough he lens o he commogni i e
amewo k, as well as g aph heo y educa ion h ough he an Hiele amewo k.
Ve ónica Ma ín-Molina is an associa e p o esso o ma hema ics educa ion a he Uni e sidad
de Se illa, Spain. She is a ma hema ician and has a Ph.D. in ma hema ics. She is cu en ly
esea ching ma hema ics lea ning h ough he lens o he commogni i e amewo k, as well as
ma hema ics eache s’ p ac ices and ma hema icians’ esea ch p ac ices.
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Isabel Ca o-To ó is a p ima y school English eache . A e ob aining he unde g adua e deg ee
in p ima y educa ion a he Uni e sidad de Se illa, she s udied a mas e in bilingual educa ion
and she is now in es iga ing he ABN me hod wi h p ima y school kids.
José Ma ía Ga ilán-Izquie do is an associa e p o esso o ma hema ics educa ion a he
Uni e sidad de Se illa, Spain. He is a ma hema ician and has a Ph.D. in ma hema ics educa ion.
He is cu en ly esea ching ma hema ics lea ning h ough he lens o he commogni i e
amewo k, as well as ma hema ics eache s’ p ac ices and ma hema icians’ esea ch p ac ices.
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Spanish p ima y school s uden s’ engagemen wi h a non- adi ional
me hod when adding and sub ac ing: Ri ualised e sus explo a o y
pa icipa ion
Ou wo k in es iga ed how six p ima y school s uden s used a non- adi ional
me hod o adding and sub ac ing: he ABN me hod, a Spanish ac onym o
Open (me hod) Based on Numbe s. Commogni i e heo y (S a d 2008) was
employed o s udy he s uden s’ ma hema ical ou ines. In pa icula , we s udied
whe he hei ou ines we e explo a o y o i ualised and ound ha ou s uden s
had a i ualised use o he ABN me hod, while wo o he s showed some signs o
incipien explo a o y use (mani es ed as an inc ease in lexibili y, applicabili y,
pe o me ’s agen i i y, and subs an iabili y). The esul s show some o he
p oblems ha s uden s ha e when applying his me hod.
Keywo ds: ABN me hod; algo i hms; commogni ion; discou se; p ima y
educa ion
In oduc ion
Algo i hms play a ele an ole in ma hema ics since hey a e linked o p oblem sol ing
(Mölle and Collignon 2019; Rasmussen e al. 2005). The i s encoun e wi h
algo i hms usually occu s in p ima y educa ion, whe e an impo an pa o he
cu iculum consis s o eaching algo i hms o add, sub ac , mul iply, and di ide
(Mo ow and Kenney 1998). These algo i hms a e ools o sol e p oblems in daily li e
and hey cons i u e he i s s ep owa ds awa eness ega ding he ma hema ical p ac ice
o building algo i hms (algo i hma ising, in e ms o Rasmussen e al. 2005).
Despi e hei impo ance in he cu iculum, algo i hms a e some imes augh
inadequa ely. Fo ins ance, Plunke (1979) poin ed ou ha con en ional ( adi ional)
algo i hms a e equen ly lea n by hea wi hou knowing he ma hema ical p inciples
and p ope ies ha suppo hem. Yea s la e , Fuson and B ia s (1990) s a ed ha many
child en ca y ou algo i hms co ec ly, bu hey canno explain why hey wo k. La e ,
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Thomas (2008) s a ed ha he s ep-by-s ep na u e o algo i hms lea es no oom o
de ia ion and hus canno os e e sa ile hinking. Mo e ecen ly, Laudano, To o iello,
and Vincenzi (2020) added ha , i eache s each only he pu ely p ocedu al aspec s o
algo i hms, hen hey un he isk o o e ing a educ ionis and dis o ed iew o
ma hema ics, which would os e disa ec ion owa ds his discipline.
The Na ional Council o Teache s o Ma hema ics (NCTM 2000) highligh ed
he need o employ complemen a y o al e na i e me hods o adi ional algo i hms, and
he e a e se e al s udies ha s a e ha employing hese me hods is mo e bene icial han
using he s anda d algo i hms. Fo example, Ulu and Özdemi (2018) concluded ha
s uden s who used algo i hmic s a egies o es ima e esul s in a i hme ic ope a ions had
less app oxima e esul s han hose who used o he s a egies, which ea i med he
hesis o Kamii and Dominick (1997). O he ma hema ics educa ion esea che s (e.g.,
Ashlock 2010; Ca pen e e al. 1998; Thompson 1998) ound ha in en ed me hods
imp o ed sel -con idence, a ou ed c i ical assessmen , and p oduced a g ea e
de elopmen o numbe sense. Fu he mo e, Neme h e al. (2019) compa ed he
lea ning ha occu ed when he e we e di e en s a egies o choose o sol e an
a i hme ic p oblem wi h he lea ning ha occu ed when ha a ie y was una ailable. In
he i s case, he e we e mo e gains in lexibili y (use o mul iple s a egies) and
adap i i y (making app op ia e s a egy choices), in he sense o Ve scha el e al.
(2009). Indeed, se e al au ho s highligh lexibili y and adap i i y in he use o
s a egies as desi able aspec s in he eaching and lea ning o elemen a y school
a i hme ic (Ba oody 2003; Ca pen e e al. 1998; NCTM 2000; To beyns and
Ve scha el 2016; Ulu and Özdemi 2018; Ve scha el e al. 2009).
In Spain, he p ima y educa ion s age (6-12 yea s) is compulso y. I is p eceded
by childhood educa ion (0-6 yea s), which is olun a y, and ollowed by seconda y
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educa ion (12-16 yea s), which is he las compulso y educa ional s age. The na ional
and egional cu icula o p ima y educa ion con ain a “numbe s and ope a ions” sec ion
which emphasises ha he cons uc ion o he di e en classes o numbe s and he
decimal sys em should be con ex ualized. Mo eo e , his cons uc ion should be done
wi h he help o manipula i es in he i s s ages and p og essi ely e ol e in o abs ac
hinking, in pa allel and ela ed o a i hme ic ope a ions. The cu icula also men ion ha
he s uden s should be able o do ope a ions wi h pencil and pape algo i hms, wi h he
calcula o and men ally, choosing he bes s a egy in each case.
One o he al e na i e me hods o he use o con en ional algo i hms ha is
gaining popula i y in Spain is he ‘Open (me hod) Based on Numbe s’ p oposed by
Ma ínez Mon e o (2011), called ABN o he Spanish ‘Abie o Basado en Núme os’.
Acco ding o Ma ínez Mon e o (2011), s uden s who mas e he ABN me hod lea n o
calcula e men ally a a high le el, sol e p oblems mo e e ec i ely, and can explain
wha hey a e doing. Fu he mo e, he indica es ha his me hod is also bene icial in he
long e m, since he ound ha s uden s who used i in p ima y educa ion ended o ha e
ewe concep ual e o s in seconda y educa ion.
The e ha e been se e al s udies ocusing on examining he di e ences in
ma hema ical compe ence o abili ies be ween p ima y s uden s who we e augh only
he ABN me hod and s uden s who we e augh only he adi ional algo i hms
(Adamuz-Po edano and B acho-López 2014; A agón-Mendizábal e al. 2017; B acho-
López e al. 2014; Ce da e al. 2018; Ma ínez Mon e o 2011; Mendoza Velazco e al.
2020). The esul s o all hese s udies ound ha he i s showed a highe le el o
ma hema ical compe ence. Howe e , none o hese wo ks in es iga ed quali a i ely how
he s uden s use he ABN me hod o hei explana ions o why hey ca y ou each s ep.

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Ou objec i e is o explo e, h ough he commogni i e amewo k, how p ima y
school s uden s ha e lea ned o use he ABN me hod. The a icle is o ganised as
ollows: we i s p esen he ABN me hod and hen in oduce he heo e ical amewo k
used o analyse ou da a. Nex , we p esen he me hodology o he s udy, he esul s, and
he sec ion o conclusions and discussion.
ABN me hod
The ABN me hod was de ised by he eache and esea che Jaime Ma ínez
Mon e o (2011) as an al e na i e way o sol ing p oblems wi hou ecu ing o he
adi ional closed me hods based on digi s. The Spanish ac onym ABN s ands o Open
(me hod) Based on Numbe s. I is called open because he e is ee choice in each s ep
o he a i hme ic ope a ion and based on numbe s because, in each s ep, comple e
numbe s a e manipula ed (and no jus digi s as in he case o adi ional algo i hms).
The e o e, he ABN me hod is no an algo i hm in he adi ional sense, which Thomas
(2020) de ines as a s ep-by-s ep se o ins uc ions a anged in logical o de o sol e a
speci ic ask. Since ABN is an open me hod, Ma ínez Mon e o (2011) claims ha i
adap s o di e en lea ning paces because each s uden can ca y ou he s eps hey need
and in he way ha bes sui s hem. Mo eo e , he ABN me hod is always in oduced in
he con ex o sol ing wo d p oblems de i ed om eal si ua ions, which allows he
s uden o associa e he a i hme ic ope a ion wi h he ac ion o ela ion exp essed in he
wo d p oblem.
This wo k ocuses on p oblems ha can be sol ed by adding o sub ac ing.
Se e al examples o addi ion and sub ac ion by he ABN me hod a e shown below.
Since he ABN me hod is no an algo i hm, he e a e se e al ways o pe o m an
addi ion. As an example, we show how o compu e 645+178 in wo ways. The smalles
addend (178) can be di ided in o pa s (as many as he s uden wishes) o be added o
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645 in successi e s eps. This me hod uses he ac ha 645+178=(645+x)+(178-x) o
any numbe x. The calcula ions a e usually w i en in a g id wi h h ee columns
(no mally wi hou labels), like he one in Table 1. The i s column indica es he
quan i y ha is going o be added, he second one has he pa ial sums, and he hi d one
he quan i ies ha emain o be added. When he e is a ze o in he hi d column, he las
numbe in he second column is he sum.
Table 1. Example o addi ion using ABN. The numbe in bold indica es he inal esul .
645+178
100
745
78
5
750
73
50
800
23
23
823
0
The example in Table 1 ook ou s eps. In he i s s ep, 100 has been added,
lea ing as a pa ial esul in he second column 745 (= 645+100) and in he hi d column
78 (= 178-100). This emo ed all he hund eds om he second addend. Wi h he sums
o he second and hi d s eps (745+5=750 and 750+50=800), he e is 800 as a pa ial
esul in he second column, and hen i is no di icul o add he emaining 23, which
amoun s o a o al o 823.
When using he abo e me hod, each s uden can decide which quan i ies o add
in each s ep and in wha o de . The e o e, he e a e o he possible g ids, each one wi h
as many ows as s eps a e needed. Fo ins ance, Table 2 shows how o add 645+178 in
h ee s eps ins ead o he ou s eps o Table 1. The di e ence is ha , in he g id o
Table 2, he second and hi d s eps o he p e ious g id ha e been me ged in a single
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s ep. This g id would be app op ia e o mo e ad anced s uden s since some o i s s eps
would be mo e di icul .
Table 2. Ano he example o addi ion using ABN. The numbe in bold indica es he
inal esul .
The e a e also a ious me hods o pe o ming sub ac ion. In he ollowing, we
show how o compu e 335-246 in se e al ways. The i s way would be by successi e
sub ac ions, ha is, by sub ac ing he same quan i ies om bo h minuend and
sub ahend un il one o hem disappea s. The emaining numbe would be he esul .
This me hod uses he ac ha 335-246=(335-x)-(246-x) o any numbe x. The
calcula ions a e also w i en in a g id wi h h ee columns, like he one in Table 3. The
i s column in his case indica es he quan i y ha is going o be sub ac ed om bo h
minuend and sub ahend, and he second and hi d columns ha e he minuend and
sub ahend ha a e becoming smalle and smalle un il he sub ahend is ze o. When
ha happens, he las minuend is he di e ence.
Table 3. Example o sub ac ion using ABN. The numbe in bold indica es he inal
esul .
645+178
100
745
78
55
800
23
23
823
0
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The example in Table 3 ook i e s eps. In he i s s ep, 200 has been
sub ac ed, lea ing as a pa ial esul in he second column 135 (= 335-200) and in he
hi d column 46 (= 246-200). This emo ed all he hund eds om he sub ahend. In he
second and hi d s eps, he ens and uni s a e emo ed om he minuend, leading o he
sub ac ion 100-11. To inish, a en is i s emo ed and inally jus one.
Each s uden can choose he quan i ies o sub ac om he minuend and
sub ahend. Fo ins ance, Table 4 shows how o do he sub ac ion 335-246 in ou s eps
ins ead o he i e s eps o Table 3. The di e ence wi h he g id o Table 3 is ha he
second and hi d s eps o ha g id ha e been me ged in one s ep. S uden s can also
eo de he s eps depending on he o de in which hey sub ac uni s, ens, hund eds,
e c.
Table 4. Ano he example o sub ac ion using ABN. The numbe in bold indica es he
inal esul .
335 - 246
200
135
46
35
100
11
335 - 246
200
135
46
30
105
16
5
100
11
10
90
1
1
89
0
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The p oposed p oblems a e p esen ed below. The names o he schools, owns,
and he game console a e all blinded.
(1) The e a e 673 s uden s in SCHOOL A and 594 s uden s in SCHOOL B. They
wan o go o he San Ped o Thea e oge he . How many s uden s will go o he
hea e?
(2) The dis ance om CITY 1 o CITY 2 is 996 kilome es and he dis ance om
CITY 1 o CITY 3 is 321 kilome es. How many mo e kilome es a e he e om
CITY 1 o CITY 2 han om CITY 1 o CITY 3?
(3) Ou e ige a o b oke down and we had o buy a new one. I my a he had
sa ed 734 eu os o buy a new e ige a o and i inally cos only 372 eu os, how
much money do we ha e le ?
(4) The e we e 643 ees in he o es and 268 ha e been cu down. How many ees
a e le in he o es ?
(5) My cousin Ma ia and I a e sa ing o buy he new GAME CONSOLE because i
is on sale o 537 eu os. Ma ia has 438 eu os and I ha e 277 eu os. How much
money do we ha e oge he ? A e we sho o money o buy he CONSOLE, o
do we ha e any money le ? How much?
The i s ou p oblems can be sol ed wi h a single a i hme ic ope a ion ( he i s
wi h a sum and he second, hi d and ou h p oblems wi h a sub ac ion). The i h one
needs an addi ion and a sub ac ion.
Da a sou ces and da a
An audio eco de and a ideo came a we e used o da a collec ion. The pa icipan s’
pa en s ga e hei consen o eco d he s uden s. The audio eco dings we e comple ely
ansc ibed by one o he esea che s, and some s ill images om he ideos we e

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selec ed. To p ese e he iden i ies o he s uden s, o his wo k, hose images in which
only he g id o he manipula i es appea we e selec ed. S uden s’ w i en esponses
we e also collec ed. To acili a e eading, cla i ica ions abou ac ions ha can be seen in
he ideo ha e been included in he ansc ip s in squa e b acke s.
Analysis p ocedu e
The da a was analysed indi idually by he in es iga o s, hen join sessions we e held o
discuss disc epancies un il a consensus was eached. Speci ically, he ocus was on
iden i ying he ou ines ha each s uden used o sol e he p oblems. Those ou ines
we e in e ed by examining he ansc ip s and images o he s uden s employing isual
media o s (g ids, hund eds cha s, oo hpicks, e c.) o ind ecu ing pa e ns.
Some s uden s sol ed some o he p oblems men ally and emained silen
h oughou he esolu ion p ocess; howe e , o he s add essed he eache o he o he
s uden and used isual media o s, which helped us de e mine how hey pe o med
speci ic s eps ha a e isible on he g ids. We hen sea ched o cha ac e is ics o he
use o ou ines ( lexibili y, applicabili y, pe o me ’s agen i i y, o subs an iabili y) ha
may indica e de- i ualisa ion.
Resul s
The e we e no iceable di e ences among he ou ines ha he six s uden s used o sol e
he p oposed p oblems. While ou o hem (Anna, Daisy, Elsa, and Flo a) seemed o
engage wi h he ABN me hod as i pe o ming a i ual, wo o he s (Be h and Ca la)
appea ed o ha e pa ially unde gone a p ocess o de- i ualisa ion because hey had
mo e lexibili y when using a ou ine, hey we e able o choose he mos app op ia e
ou ine wi hou he eache ’s help and hey could explain he s eps o hei p ocedu e.
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To be e illus a e hese di e ences in pe o mance among he s uden s, we will
highligh he cases o wo s uden s, one whose engagemen wi h he me hod ook he
o m o a i ual (Flo a) and one whose explo a ions showed ha she was in he p ocess
o de- i ualising he use o he ABN me hod (Ca la).
Flo a’s use o he ABN me hod was i ualised
Flo a always ied o use he same p ocedu e when doing he compu a ions:
decomposing he smalles numbe in o uni s, ens, and hund eds and hen adding o
sub ac ing hose uni s, ens, and hund eds. Fo ins ance, when sol ing he ou h
p oblem, she decided o do he sub ac ion 643-268 by decomposing 268 as 200+60+8
and hen sub ac ing hose h ee smalle numbe s in h ee di e en s eps, as can be seen
in Figu e 2.
Figu e 2. Flo a’s g id o p oblem 4.
This c ea ed di icul ies o he in some o he pa ial sub ac ions. Fo example,
in he second s ep, when sub ac ing 60 om 443, she hesi a ed when she had o mo e
om he 400s o he 300s:
Flo a (F): Now minus 60. 43, 33, 23… 83.
P ese ice eache (PT): Then i is...
F: 683… no, 383.
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In he i s line, we see ha she coun ed backwa ds by 10 om 443. To acili a e
his, she decided o ope a e wi h only he uni s and ens, igno ing he hund eds. Ins ead
o saying ‘443, 433, 423, …, 383’, she i s said ‘43, 33, 23…83’, which caused he
p oblems la e when she had o ‘ eco e ’ he hund eds digi . Indeed, she i s said ‘683’
ins ead o ‘383’. She had los ack o he comple e numbe , which con adic s one o
he p inciples o he ABN me hod. Mo eo e , sub ac ing 60 in he second s ep is no
he mos e icien choice o beginne s in men al a i hme ic because his o ced he o
change hund eds.
Flo a epea ed his p ocedu e when sol ing p oblems 2, 3, and 5. I is wo h
men ioning ha she con used a uni wi h a en in p oblem 2 (she says ha 676-1 = 666),
al hough she used he hund eds cha as a isual media o on se e al occasions.
We can deduce ha he use o he ABN me hod was i ualised because she used
he same p ocedu es in all he p oblems, e en when hey we e no e icien . The e o e,
he use o ou ines was no lexible.
The only p oblem she sol ed di e en ly was he i s one, whe e she used a
di e en ou ine due o he sugges ion o he eache . In ha p oblem, which can be
sol ed by adding 673 and 594, she began he same way, by decomposing 594 as
500+90+4. Howe e , in he second s ep, when ying o add 90+1173, his s uden said:
F: 11 plus 9 a e 19, a en’ hey? [She wan s o add 11, which a e hund eds in he
numbe 1173, and 9, which a e ens in 93]
The eache hen sugges ed ha Flo a use he oo hpicks as a isual media o o
ep esen he addends. Howe e , Flo a was unable o add hose numbe s in his way,
e en using he hund eds cha (Figu e 3). The eache inally p oposed:
PT: Why don’ you ake smalle numbe s ins ead o 90, o example, 20 by 20, o
how much is i om 73 o one hund ed?
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F: I don’ know…
PT: Le ’s ake a smalle numbe . How much is i o one hund ed?
Flo a also had some p oblems wi h his s ep (bo h wi h 1173+27 and wi h 94-
27), bu wi h he help o he eache and he hund eds cha managed o ge he esul
(see Figu e 4).
Figu e 3. Flo a using he hund eds cha .
Figu e 4. Flo a’s g id o p oblem 1.
Mo e e idence has been ound ha Flo a had p oblems wi h he change o
hund eds. Fo example, in he second s ep o he hi d p oblem, she had o use he
hund eds cha o 434-70. She coun ed backwa d by ens: 434, 424, … bu when she
eached 404, she again needed he help o he eache . Mo eo e , in he i h p oblem, o
do 638+70, she needed again he hund eds cha . In he same p oblem, Flo a said ha
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215-30 equals 385 and he eache had o co ec he , and also when she said ha 185-7
equals 778 ins ead o 178. In summa y, Flo a los ack o wha she was doing when
compu ing wi h la ge numbe s.
Flo a did no seem o ha e aken any s eps in he p ocess o de- i ualisa ion
because she showed no lexibili y (she always ied o employ he same p ocedu e o
sol e he asks), she had li le agen i i y (since she depended on he eache o decide
wha o do and usually needed help wi h he compu a ions), and she did no know how
o subs an ia e he choices. Fo example, when sol ing he hi d p oblem, she did no
e en know how o jus i y he choice o ope a ion:
F: Add.
PT: Tell me why.
F: Sub ac , I don’ know.
PT: Bu ell me why. I don’ mean ha i ’s w ong.
F: A sub ac ion, i is a sub ac ion, I don’ know why.
Th ee o he s uden s (Anna, Daisy, and Elsa) engaged wi h he ABN me hod in a
simila way o Flo a. Fo ins ance, on some occasions, hose s uden s con used he
columns o he g id in a s ep o he me hod, o hey did a sum when he co ec
ope a ion was a sub ac ion, o ice e sa. This way o doing ope a ions on a g id ha
has no indica ions o wha o do in each column led some o he s uden s o lose ack o
wha hey we e doing, ha is, hey did no make sense o hei pa ial addi ions o
sub ac ions. In addi ion, he e we e also si ua ions in which hose s uden s chose pa ial
ope a ions ha hey we e no able o pe o m wi hou he eache ’s help because hey
wo ked wi h e y la ge numbe s ins ead o di iding hem in o smalle pa s. In
conclusion, hei ( i ualised) use o he ABN me hod was no ad an ageous o hem,
since hey had no lea ned o use i explo a i ely o unleash i s ull po en ial.

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Ca la some imes ook ad an age o he po en ial o he ABN me hod and
adap ed he p ocedu e o he p oblem she was sol ing
None o he s uden s showed signs o ha ing comple ely de- i ualised hei ou ines
when using he ABN me hod, bu bo h Be h and Ca la had aken s eps in ha di ec ion.
In ac , Ca la some imes ied o adap he p ocedu es o he di e en calcula ions she
had o make ins ead o always applying he same sequence o ins uc ions. Some imes,
she seemed o show signs o lexibili y in he use o he ABN me hod, al hough i led
he o hesi a e when w i ing down some o he numbe s (such as 1094) o needed he
help o he eache when doing some sums (such as 1094+100). Howe e , in o he
compu a ions, she made mis akes ha made us ques ion i she was engaging wi h he
ABN me hod as i i we e a i ual.
In he i s p oblem, which can be sol ed by adding 673 and 574, he a ow ha
Ca la d ew indica es ha she added 673+574, a he han adding 574+673 as he es o
he s uden s did (Figu e 5). The e o e, she swapped wo columns o he g id. Howe e ,
when adding 673+574, she began by adding he same numbe o bo h 673 and 574,
ins ead o adding a numbe o one o hem and sub ac ing i om he o he . When she
became awa e o he mis ake, Ca la e ased he wo k un il hen and s a ed again. This
lea es some doub s abou whe he Ca la’s ou ine was p ocess-o ien ed a he han
ou come-o ien ed.
A e s a ing he compu a ion o he second ime, Ca la decided o add 673 by
decomposing ha numbe as 500+100+73. She did no explain why she decided on ha
pa icula decomposi ion, which ga e he p oblems in he i s wo s eps. In he i s
one, when adding 500+594, she co ec ly said ha i is 1094, bu she consul ed he
eache o check i she had w i en i down co ec ly. In he second one, when adding
1094 and 100, she again asked he eache o help:
23
Ca la (C): [P ese ice eache ’s name], i I pass 100, is i wo housand?
PT: No. I his is one housand plus hund ed, i is one housand and… i hese a e
hund eds and hese a e housands, how many a e one housand plus one hund ed?
C: One housand one hund ed?
PT: Ve y good!
[…]
PT: Be ca e ul, he [numbe ] 4 is missing.
Figu e 5. Ca la’s g id o p oblem 1.
Ca la o go ha she was adding 1094+100 ins ead o only 1000+100. When she
w o e 1100 ins ead o 1194, he eache wa ned he abou he missing ‘4’ bu no abou
he missing ‘9’. Ca la’s answe may ha e happened because he ABN me hod ad oca es
o compu ing using he comple e numbe s, which is mo e di icul o s uden s when
he numbe s a e big o hem.
In he es o he p oblems, Ca la did no ask o help again and showed signs o
using mo e ad anced p ocedu es. Indeed, in he hi d and ou h p oblems, she did he
sub ac ions by i s sub ac ing he maximum numbe o uni s, ens, and hund eds ha
she could om bo h numbe s wi hou eg ouping. In he hi d p oblem, she sub ac ed
332, hus emo ing all he uni s and hund eds om 372 and all he ens om 734, and
hen sub ac ed he es o he sub ahend (see Figu e 6). Tha i s s ep is ce ainly e y
24
e icien , al hough no he second, which o ced he o do 402-40. When Ca la ealised
ha he pa ne , Daisy, had di icul y wi h he pa ial sub ac ion 402-40, she ied o
help he by elling he how she could do i wi h he aid o he inge s:
Daisy (D): Yes, I know how o do i , bu wi h he [hund eds] cha .
PT: Come on, hink, 402 minus 10 is...
D: h ee hund ed... and wo
PT: Remembe ha hey a e ens, no hund eds. Recall he hund eds cha .
C: I you ake away en om 400, how much is i ? Hold up 10 inge s o you
hand and hink ha he e a e 10 o hem.
This could be cha ac e ised as a u he s ep in Ca la’s agen i i y, since she no
only makes he own decisions, bu can help he classma e by adding o he eache ’s
explana ion.
In he ou h p oblem, Ca la i s sub ac ed all he hund eds om 268 and all
he uni s and ens om 643 (Figu e 7), hus showing applicabili y in ha ou ine. The
di icul y came when pe o ming some o he sub ac ions. In Figu e 7, in he i s ow
o he g id, she was sub ac ing 243 om bo h he minuend and sub ahend, so he o he
wo numbe s should be 400 and 25, ins ead o 40 and 225. Con inuing wi h he numbe s
ha she had, ins ead o sub ac ing 40 in one s ep (which would mean sub ac ing 225-
40), she decided o i s sub ac 20 and hen he o he 20.
Figu e 6. Ca la’s g id o sol e p oblem 3.
Figu e 7. Ca la’s g id o sol e p oblem 4.
25
This las choice and he choices she made in he i s s ep o bo h he hi d and
ou h p oblems show ha Ca la was able o adap he p ocedu e o he numbe s ha she
had o wo k wi h so ha he sub ac ions did no o ce he o eg oup.
Ca la’s solu ion o p oblem 5 was again a mix u e o explo a ions and i uals.
She began by inco ec ly iden i ying he a i hme ic ope a ions ha she needed o
pe o m. She said ha she needed o do an ‘addi ion-sub ac ion’, which she p oceeded
o do on a single g id (Figu e 8).
Figu e 8. Ca la’s g id o sol e p oblem 5.
In ha g id, Ca la w o e in he i s column he numbe ha she was adding o
sub ac ing (depending on he ope a ion she was doing) and in he o he h ee columns
she w o e he numbe s ha emained a e ope a ing on 537, 438, and 277. In he i s
ow o he g id, Ca la again showed ha she could adap he p ocedu e o he si ua ion
by emo ing all he uni s and ens om he minuend and all he hund eds and ens om
32
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