EURASIA Jou nal o Ma hema ics, Science and Technology Educa ion, 2019, 15(2), em1662
ISSN:1305-8223 (online)
OPEN ACCESS Resea ch Pape h ps://doi.o g/10.29333/ejms e/100640
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c uen [email protected] (*Co espondence) Edelmi [email protected] gsanchezma amo [email protected]
and ea.ca [email protected]
The Unde s anding o he De i a i e Concep in Highe Educa ion
Claudio Fuen ealba 1*, Edelmi a Badillo 2, Glo ia Sánchez-Ma amo os 3, And ea Cá camo 1
1 Facul ad de Ciencias de la Ingenie ía, Uni e sidad Aus al de Chile, Valdi ia, CHILE
2 Depa amen de Didàc ica de la Ma emà ica i de les Ciències Expe imen als, Uni e si a Au ònoma de Ba celona, Ba celona,
SPAIN
3 Depa amen o de Didác ica de las Ma emá icas, Uni e sidad de Se illa, Se illa, SPAIN
Recei ed 1 June 2018 ▪ Re ised 17 Augus 2018 ▪ Accep ed 27 Sep embe 2018
ABSTRACT
The aim o his wo k was o iden i y and cha ac e ize he le els o de elopmen o
de i a i e schema. In o de o do so, a ques ionnai e o 103 uni e si y s uden s wi h
p e ious ins uc ion in Di e en ial Calculus was applied. The ques ionnai e was
composed o h ee asks. Fo he iden i ica ion o he le els o de elopmen o schema
and hei subsequen cha ac e iza ion, we conside he amewo k p oposed by he
APOS heo y. In pa icula , his amewo k was ope a ionalized h ough he
es ablishmen o 27 a iables ha allowed o he b eakdown o he esolu ion
p o ocols om he ques ionnai e in o disc e e elemen s. In his way, we ob ained a
ec o associa ed wi h each o hese a iables. The iden i ica ion o s uden s assigned
o each le el o de elopmen o schema was ca ied ou by a clus e analysis.
Subsequen ly, we pe o med a s a is ical analysis o equencies and implica i e, wi h
he 27 a iables, which allowed o cha ac e ize he le els o de elopmen iden i ied.
Keywo ds: de i a i e schema, le els o de elopmen , clus e analysis, implica i e
analysis, calculus
INTRODUCTION
Calculus is one o he g ea es and mos impo an achie emen s o he human in ellec , and is a hallma k o he
de elopmen o ma hema ics oday, whose powe and lexibili y may be seen in i s abili y o educe complex
p oblems o ules and simple p ocedu es in he mos di e se a eas o knowledge, such as ma hema ics, physics,
enginee ing, social sciences and biology, among o he s (Be y & Nyman, 2003; Kleine , 2001).
In his sense, Fe ini-Mundy and Lau en (1994, p.120) desc ibe Calculus as: “Calculus is a c i ical landma k in
he ma hema ical p epa a ion o s uden s in ending o pu sue nea ly all a eas o science” Fo his pa , Tall (1997, p.
289) s a es ha Calculus “is bo h a climax o school ma hema ics and a ga eway o u he heo e ical
de elopmen s”, which makes i a ansi ional poin be ween elemen a y ma hema ics and ad anced ma hema ics.
Despi e he ele ance o Calculus, one p oblem ha has s ill emained un esol ed is how o achie e
unde s anding by uni e si y s uden s in he undamen als concep s o his cou se. Mo eo e , Calculus is ypically
conside ed a di icul subjec o uni e si y s uden s; i is no ed ha hese s uden s o en can sol e p oblems
in ol ing he p ope applica ion o ules o algo i hmic p ocedu es, bu none heless ha e di icul ies when hey
ha e o sol e non- ou ine p oblems in ol ing he unde s anding o concep s (Selden, Selden, Hauk, & Mason,
1999), o he applica ion o ha unde s anding o eal-wo ld p oblems (Tall, 1992).
In pa icula , his s udy ocuses on he concep o de i a i es, which is one o he cen al and s uc u al elemen s
o any Calculus cou se, and is also a undamen al ool in he s udy and unde s anding o phenomena ha in ol e
changing o a ying magni udes (V ancken & Engle , 2014). The e o e, i co esponds o a basic concep applicable
o many o he ields in uni e si y cu icula in ma hema ics, enginee ing, and o he sciences.
Despi e he impo ance o he de i a i e concep , he esul s o esea ch ela ed o unde s anding hey ind ha
his is e y complex, showing a signi ican amoun o uni e si y s uden s who only manages o achie e a pa ial
Fuen ealba e al. / Le els o De elopmen o he De i a i e Schema
2 / 15
unde s anding o his concep (Asiala e al., 1997; Bake , Cooley & T igue os, 2000; Cooley, T igue os & Bake , 2007;
Fe ini-Mundy & G aham, 1994; Fuen ealba, Sánchez-Ma amo os, Badillo & T igue os, 2017; O on, 1983; Sánchez-
Ma amo os, 2004; Sánchez-Ma amo os, Ga cia & Llina es, 2006, 2008; Whi e & Mi chelmo e, 1996).
An aspec ha has caused di icul ies in he unde s anding o he de i a i e concep , ela es o he use o
eaching p ac ices ha ha e a o ed he lea ning o algo i hmic me hods by s uden s (A igue, 1995). This
p edilec ion o eaching algo i hmic p ocedu es makes s uden s show se ious di icul ies and e o s when aced
wi h sol ing asks ha equi e unde s anding he meaning o he de i a i e, ei he h ough i s analy ical
exp ession, as he limi o inc emen al a io, o o i s geome ical in e p e a ion as slope o he angen line (Bake
e al., 2000; Cooley e al., 2007; Sánchez-Ma amo os e al., 2006; Sánchez-Ma amo os e al., 2008).
The p oblem o unde s anding o he de i a i e concep , hough no new, is s ill one o he bigges challenges
o ma hema ics educa ion a he uni e si y le el, and is a cons an conce n o ins i u ions o highe educa ion as i
leads o low g ades, high a es o ailu e, and he abandonmen in Calculus cou ses (B essoud, Mesa, & Rasmussen,
2015; Fe ini-Mundy & G aham, 1991). Conside ing his si ua ion and he social demand conce ning esea ch in
ma hema ics educa ion, no only analyzing he p oblems o he eaching and lea ning o he discipline, bu also
con ibu ing o he solu ion, ou ocus in his esea ch is o de e mine he ollowing: how is he unde s anding o
he de i a i e concep is de eloped in uni e si y s uden s wi h p io ins uc ion in Di e en ial Calculus? Wha a e
he cha ac e is ics o he di e en le els o de elopmen o de i a i e schema? Thus, we in end o deepen he
unde s anding o how he concep o de i a i es is o med in he minds o s uden s in o de o his in o ma ion o
con ibu e in he u u e o sol ing he p oblems associa ed wi h lea ning.
BACKGROUND AND THEORETICAL FOUNDATIONS
The unde s anding o concep s is an impo an goal o ma hema ics educa ion, and i is gene ally accep ed ha
his occu s when he pieces o knowledge ha make up a concep a e men ally connec ed oge he (Hiebe &
Ca pen e , 1992). As s uden s ace new si ua ions and expe iences, hey in e ac wi h you exis ing knowledge ( on
Glase s eld, 1983), hus allowing such connec ions a e made. New ideas can be in eg a ed in o exis ing schema o
mindse s o he s uden s, and can make he exis ing knowledge econnec in new ways (Siegle , 1986).
Unde s anding is seen in his way as a connec ed ne wo k, and pieces o knowledge ha a e buil o e ime h ough
he in e ac ion o p io knowledge wi h new in o ma ion.
In pa icula , his wo k has been conside ed a cogni i e app oach ha , like he abo e ideas, conside he
unde s anding o a ma hema ical concep as a g adual p ocess o building and ela ionship be ween cogni i e
s uc u es. As i s epis emological e e ence, his app oach akes cons uc i is ideas based on he gene ic
psychology o Piage , which ha e been conside ed by Dubinsky and a g oup o esea che s known as “Resea ch
on Unde g adua e Ma hema ics Educa ion” (RUMEC), who ha e de eloped a heo e ical amewo k known as
APOS heo y (ac ion-p ocess-objec -schema). This amewo k is he esul o he in e p e a ion o Piage ian ideas
conce ning e lec i e abs ac ion.
To unde s and how he APOS heo y ope a es, i is impo an o no e ha he p inciple o e lec i e abs ac ion
was conside ed by Piage as he main mechanism o all men al cons uc ion, as well as he mechanism h ough
which all logical-ma hema ical s uc u es can de elop in he mind o an indi idual (A non e al., 2014). Acco ding
o Piage , his p inciple has wo pa s:
The i s pa in ol es e lec ion, in he sense o awa eness and con empla i e hough , abou wha Piage called
con en and ope a ions ha con en , and in he sense o e lec ing con en and ope a ions om a lowe cogni i e
le el o s age o a highe one [...]. The second pa consis s o he econs uc ion and eo ganiza ion o he con en
and ope a ions on his highe s age, ha esul s in he ope a ions hemsel es becoming con en , o which new
ope a ions can be applied (Piage , 1973; quo ed in A non e al., 2014, p. 6)
In he APOS heo y, ac ions, p ocesses, objec s and schema a e men al s uc u es ha , acco ding o his amewo k,
an indi idual cons uc s when lea ning a pa icula ma hema ical concep , while he passage h ough hese s ages
is no necessa ily sequen ial (T igue os, 2005). The mechanism o mo e om one s a e o cons uc ion o
ma hema ical knowledge o ano he , in his heo y, is e lec i e abs ac ion, which is a men al ool o de ice used in
Con ibu ion o his pape o he li e a u e
• This esea ch desc ibes he quan i a i e analysis o da a ha a e gene ally app oached by quali a i e
me hods.
• The esul s o his esea ch con i m and ex end conclusions ob ained h ough quali a i e s udies on he
de elopmen o he de i a i e schema.
• The me hodological design p oposed in his esea ch can se e as a model o he analysis o schemas o
di e en ma hema ical concep s.
EURASIA J Ma h Sci and Tech Ed
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he p ocess o knowledge cons uc ion, which allows he s uden o in e hei p ope ies on he basis o ac ions on
objec s, o o in e he ela ionships be ween objec s o he same le el o hinking. This implies, among o he hings,
he o ganiza ion o in o ma ion in an in ellec ual amewo k o ganized highe -le el (Dubinsky, 1991).
One o he hypo heses on which he APOS heo y is based is ha he cons uc ion o a new concep is based on
he ans o ma ion o p e ious concep s, o which eason hese concep s should ini ially be pe cei ed by he
indi idual as objec s. The e o e, an ac ion is a ans o ma ion o objec s (p e iously cons uc ed), pe cei ed by he
indi idual and ex e nal, in he sense ha each s ep o he ans o ma ion equi es explici a ge ing and also needs
an ex e nal s imulus o execu e (A non e al., 2014). The s uc u e o ac ion is conside ed as he simples wi hin he
APOE heo y, bu no less impo an , because i is essen ial in he cons uc ion o any ma hema ical concep . Also,
a p ocess is conside ed an in e nalized ac ion, ha is o say “men al”; in which he indi idual is awa e and has
con ol o e he ans o ma ion p oduced by he ac ion. This is cha ac e ized by he abili y o imagine, skip, o
e e se he s eps in ol ed in his ans o ma ion, wi hou he need o an ex e nal s imulus. In e io iza ion is he
mechanism ha allows o he change o s uc u e, om ac ion o p ocess, which is accomplished by epea ing
e lec ion on he ac ions (A non e al., 2014). A p ocess can no only be gene a ed by he in e io iza ion o ac ions,
bu hey can also be cons uc ed om he coo dina ion and e e sal mechanisms o p ocesses. When an indi idual
is awa e o he p ocess and is able o concei e how i can be ans o med as a whole by applying ac ions o p ocesses,
i is hen said ha he p ocess has been encapsula ed in a cogni i e objec (A non e al., 2014; Asiala e al., 1996). The
mechanism associa ed wi h his change o s a e o he p ocess is called encapsula ion. Ne e heless, once a p ocess is
encapsula ed in an objec , i he indi idual is equi ed o e u n o he p ocess ha ga e ise o he objec , his can be
done by he de-encapsula ion mechanism.
Finally, he las cogni i e s uc u e p oposed by APOS heo y is called a schema. A schema o a ma hema ical
concep in pa icula is a cohe en collec ion o ac ions, p ocesses, objec s and e en o he schemas and hei
in e ela ionships, g ouped consciously o unconsciously in he mind o an indi idual, which can be used in he
solu ion o a si ua ion o ma hema ical p oblem in ol ing he concep in ques ion. The cohe ence o he schema is
e e ed o as he indi idual’s abili y o ecognize si ua ions in which he schema is applicable and which a e no
(T igue os, 2005).
The schema is ega ded as a dynamic and complex cogni i e s uc u e ha is cons an ly de eloping and e ol ing
as he indi idual lea ns. Al hough some imes i is gene ally hough ha his s uc u e is jus beginning o o m
once he objec s (due o p og ession o ac ion, p ocess, objec and schema) a e cons uc ed, cons uc ion may begin
e en a he ime he indi idual pe o ms ac ions.
Al hough he men al s uc u es (ac ion, p ocess, objec and schema) o an indi idual du ing o an indi idual canno
be di ec ly obse ed du ing he lea ning p ocess, hese s uc u es can be in e ed om he obse a ion on wha he
indi idual can do, o no o ace a pa icula si ua ion o ma hema ical p oblem (Dubinsky, 1991). Thus, h ough
obse a ion and analysis, i is possible o cha ac e ize he s age o cons uc ion o a speci ic ma hema ical concep
in which an indi idual may be ound. Figu e 1 shows he ela ionship be ween he mechanisms and he men al
s uc u es men ioned abo e.
T igue os (2005) indica es ha when a s uden is acing a speci ic p oblem in he ield o ma hema ics, he
s uden conju es up a schema o add ess he esolu ion. Upon e oking i , ha s uden b ings hose buil s uc u es
and ela ionships ha he has a ha ime in o play. Faced wi h he same ask, di e en s uden s can use di e en
Figu e
1.
S uc u es and men al mechanisms in ol ed in unde s anding a ma hema ical concep (based on A non e al., 2014, p.
18)
Fuen ealba e al. / Le els o De elopmen o he De i a i e Schema
4 / 15
s uc u es and di e en ela ionships be ween hem. Thus, when conside ing he ela ionships es ablished be ween
he buil s uc u es, di e en le els o de elopmen o he schema can be iden i ied in he esponses o s uden s who
mee he same ask o se o asks. To add ess and cha ac e ize hese di e ences in u he de eloping he
dynamism o he schema, APOS heo y p oposes he s udy o he de elopmen o a schema h ough he use o he
iad o In a, In e and T ans, p oposed by Piage and Ga cia (1983), which classi ies i in one o hese s ages,
depending on he le el o ela ions ha an indi idual can es ablish be ween he schema componen s and o he
cogni i e s uc u es. Following his idea, Piage and Ga cia (1983) de ine he le els o de elopmen o a schema as
ollows:
• In a: his le el is cha ac e ized by he disco e y o any ope a i e ac ion, and he pu sui o analyzing i s
a ious in e nal p ope ies o i s immedia e consequences, bu wi h a wo-pa limi a ion. Fi s , he e is no
coo dina ion o his p e-ope a ion wi h o he s in an o ganized g ouping; bu also, he in e nal analysis o
he ope a ion in ol ed is accompanied by p og essi ely co ec ed e o s and gaps in he in e ence ha can
be de i ed om i (p. 163).
• In e : once an ini ial s ep is comp ised, i is possible o deduce he ope a ions ha a e in ol ed, o coo dina e
wi h o he mo e o less simila ones o he c ea ion o sys ems ha in ol e ce ain ans o ma ions. While
he e is a new si ua ion he e, he e a e ne e heless limi a ions ha a ise om he ac ha he composi ions
a e es ic ed because hey can only p oceed wi h con iguous elemen s (p.165).
• T ans: This le el is easily de ined in e ms o he abo e as in ol ing, in addi ion o he ans o ma ions, a
syn hesis be ween hem. This syn hesis a i es a he cons uc ion o “s uc u es” (p.167).
F om his de ini ion o he iad o he le el o de elopmen o a schema, in ecen decades, a ious s udies ha e
been made conce ning he de elopmen o schemas. Fo example, esea ch by Bake e al. (2000) analyzed he
calculus g aphing schema h ough he solu ions gi en by s uden s o a non- ou ine a g aphic p oblem, in which i
was no he exp ession o he unc ion bu a he a se o analy ical condi ions o ha unc ion ha was p esen ed.
In hei s udy, hey posi ed ha he Calculus g aphic schema was o med by he in e ac ion o wo schemas ha hey
called “p ope y schema” and “in e al schema”. Fo his pa , Badillo, Azcá a e and Fon (2011) uses an idea simila
o ha posed by Bake e al. (2000) o an analysis o he unde s anding o he concep o de i a i es in a g oup o
i e ma hema ics and physics eache s om Colombia. The idea o coo dina ion o schemas used in hei esea ch,
conside ing ha he de i a i e schema was o med by coo dina ing he “algeb aic schema” and “g aphic schema”.
Fo he speci ic case o his s udy, e e ence was made o wo k by Sánchez-Ma amo os (2004), in which an
analysis o de i a i e schema in e ms o logical ela ionships (conjunc ion, con aposi i e, and equi alence) ha
s uden s es ablish be ween di e en ma hema ical elemen s when sol ing p oblems. The e o e, consis en wi h he
s udy by Sánchez-Ma amo os (2004), i is unde s ood ha a ma hema ical elemen is “ he p oduc o he
dissocia ion o seg ega ion o he concep ela ing o he concep and i s p ope ies” (Piage , 1973, p. 72). F om his
de ini ion, i may be indica ed ha he de i a i e concep possesses s uc u al elemen s o a di e en na u e,
cha ac e ized by he modes o ep esen a ion and he cha ac e o hese elemen s. Wi h ega d o he modes o
ep esen a ion, his pape conside s ha he concep o he de i a i e consis s o wo ypes o elemen s: analy ical
and g aphic elemen s. Also, ega ding hei cha ac e o na u e, i conside s elemen s o a speci ic na u e, i hese
elemen s e e o a speci ic p ope y a a poin , o apply o e all, as app op ia e, o a ange co esponding o a
p ope y. Thus, i is unde s ood ha a schema co esponds o he ma hema ical s uc u e o med by he
ma hema ical elemen s and logical ela ionships es ablished be ween hem, and ha can be e oked o sol ing a
p oblem (Sánchez-Ma amo os, 2004). F om he iden i ica ion o ma hema ical elemen s and logical ela ionships
es ablished be ween hem, Sánchez-Ma amo os (2004) de ined, h ough a quali a i e analysis:
• In a-de i a i e le el. No logical ela ionships a e es ablished be ween he ma hema ical elemen s (ei he
g aphical o analy ical, speci ic o o e a ching), and he possible ou lines o he ela ionship ( he logical
conjunc ion ype) be ween hem was made e oneously. Use o ma hema ical elemen s in isola ion, and
some imes inco ec ly.
• In e -de i a i e le el. Logical ela ionships a e es ablished be ween he ma hema ical elemen s used, bu
wi h limi a ions, he p edomina ing using logical conjunc ion and ela e only speci ic and/o o e a ching
ma hema ical elemen s ha a e in he same mode o ep esen a ion, analy ical o g aphic. Mo e
ma hema ical elemen s a e used co ec ly han in he p e ious le el.
• T ans-de i a i e le el. Inc eases he epe oi e o he use o he logical ela ionships (and logic,
con aposi i e, logical equi alence) be ween ma hema ical elemen s. A his le el, he “syn hesis” o modes
o ep esen a ion occu s. This leads o he cons uc ion o he ma hema ical s uc u e. (p. 73-74)
Rega ding he syn hesis po ion o he desc ip ion o he le el o de elopmen , his applies o si ua ions in which
i is necessa y o ela e (make logical connec ions) wi h g aphic and analy ical in o ma ion, ha is, using
in o ma ion om bo h imaging sys ems o conside oge he and a i e a a “ hing” ha was no known.
“Conside ing he in o ma ion oge he ” means es ablishing some so o logical ela ionship be ween ma hema ical
EURASIA J Ma h Sci and Tech Ed
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elemen s o make a decision conce ning he si ua ion ha cu en ly exis s (Sánchez-Ma amo os, 2004). In his pape ,
he desc ip o s o each le el o de elopmen based on ma hema ical elemen s and logical ela ionships o de ining
he s udy a iables p esen ed in he s udy design we e conside ed.
Also, ano he impo an aspec in he de elopmen o schemas ela ed o he issues aised by Piage and Ga cia
(1983) on he g adual g ow h o he schema and he na u e o he iad. In pa icula , hese au ho s poin ou ha :
The na u e o he elemen s o he iad is unc ional a he han s uc u al. The e o e, hey ollow a necessa y
o de , since he de elopmen o T ans, as a sys em o ans o ma ions oge he in en i ely new p ope ies, in ol es
he o ma ion o some o hese ans o ma ions a he In e le el, and ha his la e g oup in ol es he knowledge
o cha ac e s analyzed in he In a (p. 171).
This las idea has been adap ed by he APOS heo y o analyze how a schema o a speci ic ma hema ical concep
is de eloped, which o he case o his s udy co esponds o he de i a i e, in addi ion, allows o cha ac e ize he
di e en le els o schema de elopmen , and how hese ela e o each o he , which co espond o he goals o his
esea ch.
METHODOLOGICAL DESIGN
This wo k is pa o a wide in es iga ion; whose pu pose is o analyze he de i a i e schema in college s uden s
wi h p io ins uc ion in Di e en ial Calculus. The me hodological app oach adop ed is o mixed ype wi h
explo a o y and con i ma o y cha ac e , we wo ked wi h quali a i e da a and s a is ical analysis (Rocco, Bliss,
Gallaghe & Pe ez-P ado, 2003). In his pa o he wo k, we ha e ocused on quan i a i e analysis.
Pa icipan s
The pa icipan s in his s udy we e 103 college s uden s om he academic yea s 2014/2015 and 2015/2016,
wi h double majo s in Ma hema ics and Physics a a public uni e si y in he p o ince o Ba celona. All s uden s
had aken and passed a leas one subjec which included he opics o Di e en ial Calculus. The op ion o choosing
s uden s who had al eady comple ed one o mo e cou ses in Di e en ial Calculus is in en ional and is based on
wo aspec s: (1) ou in e es in cha ac e izing he le els o de elopmen o he de i a i e schema, eached by college
s uden s a e he ins uc ional p ocess; and, (2) he di icul y associa ed wi h he cha ac e iza ion o he
encapsula ion mechanism o p ocesses, demons a ed in p e ious s udies (Cooley e al., 2007; Ga cía e al., 2011;
Fon , T igue os, Badillo & Rubio, 2016; Fuen ealba e al., 2017; Sánchez-Ma amo os, 2004).
Wi h ega d o he cha ac e is ics o he pa icipan s, i may be men ioned ha he e is g ea a iabili y in hei
age ange, p io aining, and academic le el.
Ins umen
The ins umen used (see Figu e 2) and discussed in his a icle is a ques ionnai e ha was cons uc ed by
adap ing h ee asks used in p e ious esea ch on he de i a i e concep (Bake e al., 2000; Cooley e al., 2007;
Fuen ealba, Sánchez-Ma amo os & Badillo, 2015; Ga cía e al., 2011; Sánchez-Ma amo os, 2004; Sánchez-Ma amo os
e al., 2006). Fo i o be sol ed, i was necessa y o use dis inc ma hema ical elemen s ha make up he de i a i e
concep . This ques ionnai e was adminis e ed o 103 pa icipan s in his s udy and las ed app oxima ely 90
minu es.

Fuen ealba e al. / Le els o De elopmen o he De i a i e Schema
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Analysis Me hods
As an ini ial s ep be o e pe o ming he analysis, he esolu ions p o ocols we e disc e ized. Fo he disc e ize
o hese p o ocols, and o ob ain a ec o associa ed wi h each o hem, 27 a iables we e de ined (see Figu e 3).
These a iables a e he esul o he b eakdown o ma hema ical elemen s in bo h modes o ep esen a ion
Figu e 2. Tasks p oposed in he ques ionnai e and desc ip ion o aspec s associa ed wi h i s solu ion
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(analy ical and g aphical), he use o logical ela ionships and p e ious s udies using he APOS heo y (Fuen ealba
e al., 2017; Fuen ealba, Badillo & Sánchez-Ma amo os, in p ess; T igue os & Escandón, 2008). Fo example, he
decomposi ion o he logic ela ionship o double implica ion be ween he posi i e sign o he i s de i a i e in an
in e al and he s ic g ow h o he unc ion in said in e al, allowed us o gene a e he a iables V11 and V12. Also,
by decomposing o some ma hema ical elemen s in bo h modes o ep esen a ion, we gene a ed o he a iables, o
example, he a iables V9 and V10, associa ed wi h co ec use o he meaning o in lec ion poin .
Figu e 3. Va iables used o he disc e iza ion o he esolu ion p o ocols o each o he ques ionnai es
Fuen ealba e al. / Le els o De elopmen o he De i a i e Schema
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The pu pose o es ablishing hese a iables was o conduc a clus e analysis ha would iden i y and
cha ac e ize he le els o de elopmen o he de i a i e schema (g oups deli e ed by conglome a es). Howe e , o
quan i y he p esence o absence o each o he a iables in he esolu ion p o ocols used by he s uden s, i was
necessa y o use a measu emen scale o assign a sco e o each o hem. Fo he speci ic case o his s udy, a scale o
a bina y ype, 1 o 0 (1 in he case ha he co ec use o a iables was obse ed, and 0 in ano he case). F om hese
wo ools ( a iables and scale) we ob ained o each esolu ion p o ocol a ec o o he ype (V1, V2, V3,… V27),
whe ein each a iable has a alue o 0 o 1. Each o hese ec o s was labeled wi h he le e E and a subsc ip ha
indica es he s uden belonging o i , hus 103 ec o s (E1, E2, E3,. .., E103) we e ob ained.
La e , a e h ee sub-ma ices o ce ain da a (In a, In e and T ans) we e ob ained h ough a clus e analysis,
a equency analysis was conduc ed wi h pe cen age equencies in e ms o he p ope o imp ope use o he
a iables. Howe e , his ype o analysis does no indica e wha he mos impo an a iables a e wi hin each le el
o de elopmen , and also allows o he ela ionships o be iewed be ween hem. The e o e, o obse e he
unde lying s uc u es o he g oup o a iables, an implica i e s a is ical analysis (Analyze S a is ique Implica i e, o
ASI, in F ench) is ca ied ou on each o he le els, co esponding o a me hod o analysis which allows, om a se
o da a which in e ela es wi h a popula ion o subjec s o objec s wi h a se o a iables, o he ex ac ing and
s uc u ing o knowledge in he o m o ules and gene alized ules (Zamo a, G ego i & O ús, 2009). In pa icula ,
an implica i e s a is ical analysis is a me hod o da a mining, which is no symme ical, and which allows o
s a is ically model he quasi-implica ion 𝑎𝑎 ⟹ 𝑏𝑏, ha is o say, i a emp s o quan i y how likely i is o happen i
he a iable b has been obse ed in he popula ion a iable E (Le man, G as & Ros am, 1981). Acco ding o
T igue os and Escandón (2008), in his me hodology, “ he implica ion 𝑎𝑎 ⟹ 𝑏𝑏 shall be admissible i he numbe o
indi iduals E ha con adic i is e y small, in p obabilis ic e ms, in ela ion o he numbe o indi iduals expec ed
unde he hypo hesis o ha ing no ela ionship. I his happens, you can say ha A, he se o obse a ions o sa is y
he ea u e, is “almos ” con ained in B, he se o obse a ions ha sa is y he cha ac e is ic b” (p. 67). Unlike da a
based symme ical analysis me hods, o example, on a dis ance o a co ela ion, he se s o ules se s ob ained by
he analysis can lead o implica i e causal hypo heses (Zamo a e al., 2009).
ANALYSIS AND RESULTS
As men ioned, he i s s ep co esponded o he embodimen o he clus e analysis wi h 103 ec o s ob ained
by he disc e iza ion o he ques ionnai es. I is impo an o no e ha clus e analysis does no p esen a single
solu ion, bu a he he esul o his depends on he cha ac e is ics o he selec ed p ocess, i.e., he dis ance used
and agglome a i e o clus e ing me hod. While his s udy is in ended o cha ac e ize he le els o de elopmen o
he de i a i e schema, i was assumed, indica ing he APOS heo y as o he le els o de elopmen o an ou line o
any ma hema ical concep a e h ee and ha unde his amewo k a e he iad In a, In e and T ans (A non e
al., 2014; Piage & Ga cia, 1983). Gi en he o egoing, i was buil wi h he In os a 2016 applica ion six di e en
conglome a es combining dis ances (Euclidean, squa ed Euclidean and Maha an) and clus e ing me hods (simple
and comple e linkage). Values o he cophene ic co ela ions o each o he six hie a chical clus e s ob ained a e
p esen ed in Table 1.
Conside ing he esul s o he six conglome a es ha we e cons uc ed and hei cophene ic co ela ions we
choose hose whose cophene ic co ela ion coe icien was highes , which in his case co esponded o 0.859, which
was selec ed. The a ionale o his decision was based on he issues aised by Sokal and Rohl (1962), who indica e
ha his a io ensu es a co ec a ing when i s alue is close o one.
As a esul o he clus e analysis wi h he squa ed Euclidean dis ance and he comple e agglome a i e linkage
me hod, he dend og am shown in Figu e 4 was ob ained.
Table 1. Hie a chical Clus e ob ained by a ying he dis ance and he ype o g ouping selec ed
Dis ance
Clus e g ouping ype
Cophene ic co ela ion
Squa ed Euclidean
Comple e linkage
0,859
Euclidean
Comple e linkage
0,747
Manha an
Comple e linkage
0,660
Squa ed Euclidean
Single linkage
0,723
Euclidean
Single linkage
0,775
Manha an
Single linkage
0,723
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The dend og am ob ained h ough he clus e analysis allowed o he da a ma ix o be di ided in o h ee sub-
ma ices, one o each le el o de elopmen o he de i a i e schema. In pa icula , in Figu e 4, he ed po ion o
he dend og am co esponds o s uden s assigned by he clus e analysis a he T ans-de i a i e le el o
de elopmen , he blue po ion co esponds o s uden s assigned o he In e -de i a i e le el, and he g een po ion
a e s uden s assigned o he In a-de i a i e le el.
The s uden s classi ied in each o he le els o de elopmen o he de i a i e schema a e p esen ed in Table 2.
The di ision o he ma ix allowed o a desc ip i e s a is ical analysis o he pe cen ages o he co ec use o
he 27 a iables o each o he le els o de elopmen o he schema. The pe cen ages o co ec use o a iables o
each le el o de elopmen a e p esen ed in Table 3.
Figu e 4. Comple e dend og am ob ained wi h squa ed Euclidean dis ance and comple e linkage
Table 2. S uden s assigned o he le els o de elopmen o he de i a i e schema
Le el
T ans-de i a i e
In e -de i a i e
In a-de i a i e
S uden s
E1, E2, E3, E6, E7, E10, E11, E12,
E14, E16, E17, E18, E19, E20, E21,
E22, E24, E25, E27, E29, E30, E31,
E32, E34, E35, E36, E37, E39, E40,
E41, E46, E47, E48, E49, E51, E54,
E57, E59, E62, E63, E64, E65, E67,
E72, E74, E80, E83, E84, E88, E89,
E91, E93, E100, E101, E102.
E4, E5, E9, E13, E15,
E23, E26, E28, E33, E38,
E42, E43, E44, E45, E52,
E53, E55, E56, E58, E60,
E66, E68, E69, E70, E71,
E73, E75, E76, E77, E78,
E81, E82, E87, E103.
E8, E50, E61,
E79, E85, E86,
E90, E92, E94
E95, E96, E97,
E98, E99.
To al
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