INTERNATIONAL ELECTRONIC JOURNAL OF MATHEMATICS EDUCATION
e-ISSN: 1306-3030. 2019, Vol. 14, No. 3, 455-
466
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A icle His o y: Recei ed 9 Janua y 2019  Re ised 20 Feb ua y 2019  Accep ed 21 Feb ua y 2019
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OPEN ACCESS
Combining Theo e ical App oaches: Socio-Didac ic-
Ma hema ical No ms and Pe spec i es in P e-se ice Seconda y
Ma hema ics Teache s’ Discou se
Rocío Toscano 1*, Vic o ia Sánchez 1, Me cedes Ga cía 1
1 Depa amen o de Didác ica de las Ma emá icas, Uni e sidad de Se illa, SPAIN
* CORRESPONDENCE: [email protected]
ABSTRACT
This s udy is pa o wide esea ch ha seeks o in es iga e he exis ence (o no ) o socio-
didac ic-ma hema ical no ms in he discou se a ises wi hin a g oup o p e-se ice seconda y
ma hema ics eache s when hey a e sol ing a didac ic-ma hema ical ask. In addi ion, we y o
analyze whe he any o hese no ms could ha e ela ionships wi h eache s’ pe spec i es ha hey
can adop in hei u u e p ac ice. The da a comes om he ansc ip ions o he audio eco dings
o he dialogues among he p e-se ice eache s when hey a e sol ing a didac ic-ma hema ical
ask in he class oom. Based on ou analysis, we ha e been able o iden i y i e socio-didac ic-
ma hema ical no ms. Th ee o hem we e in some way ela ed o he ma hema ical con en and
i s lea ning. The o he wo a e ela ed o eache s’ ole, p o iding in o ma ion abou cha ac e is ics
ha he u u e eache s associa e wi h said ole. Fu he mo e, we ha e iden i ied ea u es ela ed
o eache s’ pe spec i es h ough he abo e-men ioned no ms.
Keywo ds: p e-se ice seconda y ma hema ics eache s’ discou se, socio-didac ic-ma hema ical
no ms, eache s’ pe spec i es
INTRODUCTION
The las ew yea s ha e seen a g owing numbe o s udies ela ed o he di e en no ms ha can be
iden i ied in discou se. In hese s udies, he no ms ha e been conside ed wi h di e en nuances. F om a
sociological pe spec i e, no ms o p esc ip ions ha e been speci ied as demands i hey a e shown in an ob ious
way o no ms i hey a e hidden assump ions o unde lying ideas (Biddle & Thomas, 1996). Au ho s such as
He bel-Eisenmann (2003) ha e p o ided in o ma ion abou di e en e minologies ha ha e been used wi h
espec o he wo d “no m”, acco ding o he adop ed heo e ical app oach.
F om an ini ial cogni i e pe spec i e, “wi h cons uc i ism as a guiding amewo k” (Yackel & Cobb, 1996,
p. 459), b oadened wi h a sociological pe spec i e, Yackel and Cobb (1996) ha e ocused on socio-ma hema ical
no ms, conside ing hese no ms as “no ma i e aspec s o ma hema ics discussions speci ic o s uden s’
ma hema ical ac i i y” (p. 461). They ha e shown how hese no ms egula e ma hema ical a gumen a ion and
in luence he lea ning oppo uni ies o he s uden s and he eache (Yackel & Cobb, 1996). O he au ho s
such as Ta sis and Koleza (2008) ha e ocused on social and socio-ma hema ical no ms, con empla ing hem
in he same sense as he abo e-men ioned au ho s. They ha e s udied how hese no ms a e es ablished du ing
he in e ac ions o p e-se ice eache s when hey sol e ma hema ical p oblems and hei e ec in he
p oblem-sol ing p ocess. The cons uc socio-ma hema ical no ms has been used by Go go io and Planas
(2005), wi h a hea ie social weigh han in Yackel and Cobb’s in e p e a ion, ocusing on he di e en e-
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in e p e a ions o he same no ms in mul ie hnic class ooms whe e he immig an s uden s el ha hey
hemsel es o hei con ibu ions we e alued nega i ely. The lack o nego ia ion caused obs acles o
immig an s uden s’ pa icipa ion in he ma hema ical con e sa ions. Fu he mo e, aking in o accoun
S a d’s wo ks (2007; 2008), Sánchez and Ga cía (2014) ha e conside ed he socio-ma hema ical and
ma hema ical no ms ha a ise in he in e ac ion be ween p e-se ice p ima y eache s when hey sol e a
ma hema ical ask, inco po a ing he ma hema ical opic add essed ( he ma hema ical de ini ion) as a new
a iable.
In gene al, esea ch ela ed o no ms has p o ided new insigh s abou how hey a e es ablished du ing he
social in e ac ions gene a ed in di e en se ings (Yackel & Cobb, 1996), i s impo an ole in s uden s’
lea ning (McNeal & Simon, 2000; Yackel, Cobb, & Wood, 1991) a di e en educa ional le els and, he e o e,
he ole o he eache o de elop he no ms in class oom (Dixon, And easen, & S ephan, 2009; McClain &
Cobb, 1997; Yackel, 2002), showing i s in luence in ma hema ics eaching/lea ning p ocesses.
P ecisely he cha ac e iza ion o he eache s’ ole in he class ooms has been conside ed by o he
esea ches. In pa icula , om a concep ual amewo k based on a social cons uc i is pe spec i e, a
coo dina ion o cogni i e and social pe spec i es on knowing and coming o know, Simon and colleagues
(Simon, Tzu , Heinz, & Smi h, 2000; Tzu , Simon, Heinz, & Kinzel, 2001) ha e u ilized he e m pe spec i e
“ o pos ula e a b oad pedagogical s uc u e composed o a mul iple concep ions ha collec i ely o ganize some
aspec s o a eache ’s p ac ice” (Tzu e al., 2001, p. 228). Taking in o accoun he impo ance o iden i ica ion
o hese pe spec i es ha unde lie eache ’s p ac ice, in o he occasions we ha e conside ed he exis ence o
ela ionships be ween eache s’ pe spec i es and he ela ional a chi ec u e ha is es ablished in class ooms
(Escude o & Sánchez, 2008), and he way in which he cons uc ion mechanisms o knowledge a e modeled by
eache s in class ooms (Ga cía, Ga ilán, & Llina es, 2012). Ne e heless, how some no ms ha egula e
class oom p ac ices could encou age he de elopmen o iewpoin s o posi ions ela ed o he adop ion o a
eache ’s de e mina e pe spec i e ha e no been gi en in-dep h conside a ion. The ela ionships o some
no ms ha a e socially sha ed by u u e seconda y ma hema ics eache s and he possible pe spec i es ha
hey can gene a e become ou objec o in e es in his wo k.
CONCEPTUAL FRAMEWORK
As we ha e men ioned in he p e ious sec ion, di e en heo e ical app oaches and pe spec i es has been
combined o coo dina ed in ma hema ics educa ion esea ch (Wedege, 2009). In his s udy, we ag ee wi h Cobb
(1994) in he sense ha : “The cen al issue is hen no ha o adjudica ing a dispu e be ween opposing
pe spec i es. Ins ead, i is o explo e ways o coo dina ing cons uc i is and sociocul u al pe spec i es in
ma hema ics educa ion” (p. 13).
Taking in mind his commen , in ou esea ch, on he one hand, he way o conside ing no ms is based on
he poin o iew o S a d (2008). In he heo e ical app oach, his esea che highligh s he uni y o
communica ing and hinking (cogni ion). She sugges s he wo d commogni ion, emphasizing ha in he
discou se p oposed in hinking, cogni i e p ocesses and in e pe sonal communica ion p ocesses a e di e en
mani es a ions o he same phenomenon. S a d (2008) conside s no ms o be “me adiscu si e ules ha a e
widely endo sed and enac ed wi hin he discou se communi y” (p. 300). On he basis o his conside a ion o
no ms, wi hou minimizing he impo ance o o he no ms, he e we ocus on socio-didac ic-ma hema ical
no ms (SDMNs) ha a ise in he in e ac ion wi hin g oups o p e-se ice seconda y eache s when hey a e
sol ing a ask p oposed in a cou se o a eache educa ion p og am.
In his s udy, we y o iden i y SDMNs in p e-se ice seconda y eache s’ discou se on he basis o he
commonly accep ed cons ain s, condi ions and pa icula i ies ela ed o ma hema ics as a school subjec
ma e , i s eaching and i s lea ning. They may a ise in he in e ac ion be ween hese p e-se ice eache s
when hey a e sol ing a ask ela ed o eache s’ p o essional ac i i y in a speci ic con ex : eache educa ion.
We assume ha “ hese cons ain s and condi ions a e consequence o he con ex in which ma hema ical
p ac ices o hese s uden s ha e been de eloped” (Sánchez & Ga cía, 2014, p. 308). In addi ion o hese speci ic
no ms, we a e awa e ha o he no ms exis , bu hey a e no add essed in his pa o ou s udy.
On he o he hand, in ou concep ual amewo k, we include he abo e-men ioned no ion o pe spec i e in
he sense o Tzu e al. (2001). In pa icula , we conside he h ee di e en pe spec i es cha ac e ized by
hese au ho s: adi ional, pe cep ion-based and concep ion-based. They emphasize ha hese pe spec i es
a e hei cha ac e iza ion o eache s’ p ac ice, and no how he eache s hemsel es would desc ibe hei
p ac ices. A adi ional pe spec i e is cha ac e ized by “a passi e s ance owa d lea ning coupled wi h a
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“ha es ing” s ance owa d ma hema ics, i exis s ou side and independen o he lea ne (o knowe ) who
needs o ob ain i ” (Tzu , 2010, p. 56). In his pe spec i e, eaching is cha ac e ized by “ eache s’ a emp s o
ansmi pa icula ma hema ical ideas o s uden s” (Tzu e al., 2001, p. 247). As Tzu poin s ou , in his
pe spec i e “ eache s eel esponsible o logically o ganizing and clea ly p esen ing he ma hema ical
con en ” (Tzu , 2010, p. 56). This con en , o him, has “c ys allized” h ough millennia. A pe cep ion-based
pe spec i e conside s “ he eache ’s p ima y ole is no o di ec ly ansmi he in ended ideas o s uden s, bu
o o ches a e condi ions ha engage s uden s in ac i ely seeing and connec ing hose ideas” (Tzu e al., 2001,
p. 247). In con as o adi ional pe spec i e, a pe cep ion-based pe spec i e emphasizes on lea ning as
necessa ily an ac i e p ocess; eaching based on his pe spec i e emphasizes “s uden s coming o see
ma hema ical ideas and ela ionships h ough hei own expe ience o he ma hema ics” (Tzu e al., 2001, p.
247). Finally, in he concep ion-based pe spec i e eache s eel esponsible o engaging lea ne s in ealis ic
asks, o ien ing lea ning e lec ions h ough a eo ganiza ion o p e iously es ablished schemes (Tzu , 2010).
F om such a pe spec i e, “ma hema ics is hough o as a web o concep ions ha humans abs ac h ough
e lec ion. Lea ning is he building up and he con inual ans o ma ion o one’s concep ions. Teaching,
p omo ing in ended concep ual ad ances” (Tzu e al., 2001, p. 247).
In his a icle we ocus on how socio-didac ic-ma hema ical no ms, which a ise in he in e ac ion wi hin
g oups o p e-se ice seconda y ma hema ics eache s when hey a e sol ing a didac ic-ma hema ical ask,
could be ela ed o he dis inc aspec s ha cha ac e ize he di e en pe spec i es. The esea ch ques ions
behind his s udy a e:
• Is i possible (o no ) o iden i y socio-didac ic-ma hema ical no ms in he discou se o p e-se ice
seconda y eache s in he con ex o sol ing a didac ic-ma hema ical ask?
• Could hese no ms ha e any ela ionship wi h hei u u e eache s’ pe spec i es?
METHOD
Taking in o accoun he na u e o he esea ch ques ions posed, his s udy adop s a
quali a i e/in e p e a i e app oach, since i ies o desc ibe, decode and in e p e he meanings o e en s
happening in a pa icula social con ex . In he ollowing, we p esen he di e en subsec ions included in ou
me hodological design.
Pa icipan s and Con ex
In he pa o he esea ch epo ed he e, pa icipan s we e u u e seconda y ma hema ics eache s
en olled in he Mas e ’s Deg ee in Seconda y Educa ion Teache T aining a a la ge uni e si y in Spain and,
speci ically, in he “Ma hema ics lea ning and eaching” cou se. This Mas e ’s is a pos g adua e cou se o 60
c edi s ECTS (Eu opean C edi T ans e Sys em), ecen ly implemen ed in Spanish uni e si ies. I plays an
essen ial ole: ea u es he necessa y p o essional equi emen s ha enable an indi idual o become a eache
a seconda y school le el.
The “Ma hema ics lea ning and eaching” cou se (12 c edi s ECTS) was ocused on ma hema ics educa ion
issues. In his cou se, he uni e si y eache p oposed didac ic-ma hema ical asks in he class oom. These
asks ied o b ing he u u e eache s close o he p o essional wo k o seconda y ma hema ics eache s. Fo
ins ance, each ask could be ela ed o a p o essional ac i i y, such as in e p e ing pupils’ ma hema ical
p oduc ions, and a ma hema ical con en , o example geome y. Fo sol ing hese asks, he uni e si y
eache s p o ided some a icles om ma hema ics educa ion li e a u e. O he in o ma ion was supplied by
uni e si y eache s’ explana ions and he ma hema ical con en knowledge coming om he p e ious
backg ound o hese p e-se ice seconda y eache s (a deg ee wi h s ong ma hema ical con en ). The p e-
se ice eache s wo ked in small g oups and he g oups we e gi en o al au onomy wi h espec o hei way
o wo king. The p e-se ice eache s had o w i e a inal epo ea u ing he indings o hei analysis. They
we e used by he uni e si y eache as a means o e alua ing p e-se ice eache s’ wo k.
On he cou se, he e we e 28 p e-se ice seconda y eache s en olled and hey wo ked in se en small
g oups in wo-hou sessions pe week. The e we e wo 5-p e-se ice eache g oups, wo 3-p e-se ice eache
g oups, and h ee 4-p e-se ice eache g oups. They we e in o med o ou esea ch (cha ac e is ics, aim,
con iden iali y issues, e c.), and hey should decide o hemsel es whe he o no hey wished o pa icipa e
in he s udy.
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In pa icula , pa icipan s we e 20 o 28 p e-se ice seconda y eache s (6 emales and 14 males),
co esponding o i e o he se en g oups (namely G1, G2, e c. on ou esea ch), ha decided o pa icipa e
olun a ily in he s udy. Al hough hey came om di e en special ies, all o hem had a uni e si y deg ee
ela ed o ma hema ics o o he scien i ic special ies. These p e-se ice eache s ha e no had o mal eaching
expe iences. Thei ages anged om 23 o 35, and hey we e e hnically and acially simila .
Da a Collec ion P ocedu es
Taking in o accoun ha in he commogni i e amewo k o S a d he discou se o s uden s is “ he p incipal
objec o inqui y” (S a d, 2008, p. 276), esea che s ha e used di e en lea ning en i onmen s o access o he
discou se (Wille & Boque , 2009). The da a o ou s udy consis s mainly o he ansc ip ions o audio
eco dings o he dialogues among he p e-se ice eache s while sol ing a didac ic-ma hema ical ask
p oposed in he abo e-men ioned cou se. I was e y di e en o adi ional asks ha a e usual in he
scien i ic ields in which hese p e-se ice eache s had ob ained hei deg ees (Sánchez & Ga cía, 2009). The e
was no in e en ion on he pa o he esea che s nei he in he design no in he implemen a ion o he asks
o he cou se. We chose one o he asks p oposed by he uni e si y eache . This ask was a didac ic-
ma hema ical si ua ion ela ed o a eache ’s p o essional ac i i y (analyzing school p oblems aken om
seconda y school ex books) and a ma hema ical con en ( unc ions), including wo a icles (see Appendix).
The p ocess o sol ing his ask was he con ex in which he da a we e gene a ed.
The choice o his ask was p ecisely condi ioned by encompassing p o essional ac i i ies which a e e y
p esen in he wo k o eache s, and a ma hema ical con en ( unc ions) ha is amilia o he p e-se ice
eache s. We hough his can encou age he communica ion among hem when hey sol e he p oposed
didac ic-ma hema ical ask, p o iding an oppo uni y o a b oad exchange o iews and a o ing ou access
o p e-se ice eache s’ discou se.
The eco dings we e collec ed du ing se en sessions o wo hou s a week pe each one o i e pa icipa ing
g oups.
Da a Analysis
Once he dialogues we e ansc ibed in o w i en ex , in a i s s ep we analyzed he discou se on he basis
o wo p ope ies p oposed by S a d (2007; 2008): wo d use and na a i es. In ou s udy, we adap S a d’s
p oposal wi h espec o a discou se ela ed o ma hema ics, i s eaching and i s lea ning. Ma hema ical wo ds
we e eplaced by didac ic-ma hema ical wo ds. Examples o hese wo ds a e: modes o ep esen a ion,
ma hema ical elemen , pupils’ unde s anding and so on. In he case o na a i es, we ocused on endo sed
na a i es conside ed as se s o p oposi ions ha a e accep ed and labeled as ue by he p e-se ice seconda y
ma hema ics eache s. Some examples o endo sed na a i es a e shown in Table 1 (highligh ed in bold).
Al hough he wo ds we e p esen ed in isola ion a imes, hey we e usually inco po a ed in o he na a i es.
The h ee membe s o he esea ch eam pa icipa ed in he iden i ica ion o he wo p ope ies. The
common iden i ica ions we e assumed and he p oblema ic iden i ica ions we e discussed o be accep ed o
ejec ed. In his way, he analysis p ocess was alida ed.
In a second s ep, om he p e ious analysis, we began o iden i y in he p e-se ice eache s’ endo sed
na a i es ea u es coming om he way o conside ing ma hema ics as a subjec ma e in he school con ex .
In a hi d s ep, when di e en na a i es sha ed ea u es ela ed o a same issue, we analyzed i s meaning
in an a emp o in e no ms. In pa icula , SDMNs we e in e ed by he esea che s on he basis o di e en
ea u es linked o aspec s coming om he way o conside ing ma hema ics as a subjec ma e o be augh
and lea n .
In Table 1, we p esen a b ie example om a g oup o p e-se ice eache s (G oup 7), o show how we
in e SDMNs.
In Table 1, he successi e inclusion in he discou se o endo sed na a i es ha emphasized di e en
ea u es ela ed o he eache ’s ole lead us o iden i y he socio-didac ic-ma hema ical no m ha we ha e
named “The eache alida es he knowledge and cla i ies doub s”.
Finally, in he ou h s ep, hese no ms we e conside ed om he di e en cha ac e is ics o he
pe spec i es ha ha e been desc ibed in he concep ual amewo k o his s udy (Tzu e al., 2001; Tzu , 2010).
In ou example, we can say he gi en eache ’s ole by hese p e-se ice seconda y eache s, exp essed in his
no m, i s wi h a adi ional pe spec i e.
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FINDINGS OF THE STUDY
Based on ou analysis, we ha e been able o in e i e SDM no ms in ou s udy. The numbe o he no m
indica es he o de o appea ance in he analysis p ocess ollowed by he esea che s. O cou se, we do no
wan o say ha all he discou se i in o hese i e no ms. They a e exclusi ely he no ms ha we ha e
in e ed.
Th ee o hem (SDMN 2, SDMN 4, and SDMN 5) we e in some way ela ed o he ma hema ical con en
and i s lea ning. Fo ins ance, om he di e en endo sed na a i es iden i ied in G oups 1 and 6, we we e
able o in e he SDM no m “To each he mos common o necessa y mode o ep esen a ion is able o alues”
(SDMN 2). In Table 2, we show endo sed na a i es ha ha e been selec ed as ep esen a i e examples o
o he na a i es ha exp ess analogous ideas (column 1) and hei iden i ied ea u es (column 2) ha leads
us o in e he SDMN 2 (column 3).
Table 1. Example o he p ocess ollowed o in e SDM no ms
Examples o endo sed na a i es
Iden i ied ea u es
In e ed no m
(G7, Page 63) [Con ex : P e-se ice eache s discuss he in o ma ion p o ided in
ela ion o he ansla ions be ween modes o ep esen a ion]
1619: D:
We a e going o call him [ e e ing o he uni e si y
eache ] o see i he heo y is he …
1620: A:
Wha ?
1621: D:
Tha we a e going o call him, igh ?
1622: A:
Yes, bu i his is all clea o us, he ques ion is he mos ...
1623: M:
Bu we a e going o ask him he doub s.…
The eache con i ms
esul s and cla i ies
doub s
“The eache
alida es he
knowledge and
sol es he
doub s”
(G7, Page 67) [Con ex : P e-se ice eache s y o decide he bes p ocedu e o
sol e p oblem]
1723: A:
Sol ing an equa ion, exac ly, okay. I hink ha ’s done, igh ?
1724: M:
Now we call him [ e e ing o he uni e si y eache ] o
check. Now when he comes we show him he answe o
ques ion 1.
1725: A:
Ok.
The eache con i ms
wha has been done
(G7, Page 75) [Con ex : P e-se ice eache s ha e inished pa o he ask]
1911: A:
[add essing o he uni e si y eache as he ep esen a i e o
he g oup] We ha e done h ee p oblems i you’d ca e o
ake a look...
1912: S:
To see wha you hink …
The eache alida es
he esponse o allow
hem o con inue
(G7, Pages 81-82) [Con ex : P e-se ice eache s a e discussing he in e p e a ion
o a g aph]
2058: M:
.... he in e p e a ion, ...yes. I’ e hough abou i , i is e y
impo an bu I don’ know i i is an elemen .
2059: A:
Okay, hen pu i as a ma hema ical elemen and we can
ask him [ e e ing o he uni e si y eache ] a e wa ds
when he comes…
2060: M:
I’m going o w i e i in pencil and hen we will know
ha we ha e o ask abou i .
We can pu wha e e
and hen con i m
wi h he eache
(G7, Pages 84-85) [Con ex : P e-se ice eache s discuss whe he he p oblem hey
a e analyzing p esen s a local o global si ua ion]
2127: A:
I hink ha he p oblem is in ...
2128: M:
I would ask him [ e e ing o he uni e si y eache ]
2129: A:
I hink ha he p oblem is he solu ion. Is i global o local?
... Because you say, well yes, I know ha his is he
maximum bu because I ake a look o all…
2130: M:
Righ
2131: A:
Exac ly, bu I hink ha i is mo e o he solu ion, bu
we a e going o ask him.
2132: M:
Ok.
2133: A:
I hink ha ’s wha i is, bu well now we can ask him...
The eache guides
abou he way
o wa d

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460 h p://www.iejme.com
This SDMN 2 emphasizes he impo ance ha is gi en in ou Spanish educa ional con ex o he use o he
able as a way o ob ain a g aphical ep esen a ion, minimizing he use o ce ain p ope ies (cu ing wi h
axes, e ices) ha can allow a be e pic u e o he si ua ion.
O he endo sed na a i es iden i ied in G oup 4 lead us o he SDM no m “A ma hema ical esul is (o is
no ) co ec depending on si ua ion” (SDMN 4). In Table 3, we show some ep esen a i e examples o hese
endo sed na a i es.
The SDMN 4 indica es how a solid aining in scien i ic con en helps o link ma hema ics esul s o he
posed si ua ion, es ablishing an explici ela ionship be ween he g aphical ep esen a ion and he e bal
si ua ion. P ecisely, his solid aining can lead o he SDM no m “Explana ions in he answe s o he asks
a e no necessa y because ime is was ed” (SDMN 5), iden i ied in G oup 7 as we can see in he ep esen a i e
example shown in he Table 4.
The endo sed na a i es om which his example is ep esen a i e sugges ha , o hese u u e eache s,
communica ion is nei he conside ed a ele an ma hema ical p ocess no seems o ecognize he impo ance
o i s ole in lea ning.
The o he wo SDMNs (SDMN 1 and SDMN 3) we e linked o eaching in gene al and eache s’ ole in
pa icula . One o hem (SDMN 1: “The eache alida es he knowledge and cla i ies doub s”) was iden i ied
in all he g oups and has been de ailed in he analysis sec ion (see Table 1). We can in e ha u u e eache s
need a eache ’s p esence and opinions o alida e hei wo k. Two aspec s eme ge: Doub s a e no discussed
ho oughly in he g oups, and possible al e na i e answe s a e no discussed. Fu u e eache s hink ha
accu a e solu ions and al e na i es come om he eache . We can hink ha his is no mal p ac ice in
eaching, and no hing new has been iden i ied. Ne e heless, his show how a de e mina e way o seeing he
ma hema ical class oom ( eache -cen e ed eaching) is pe pe ua ed.
Table 2. Rep esen a i e example o all he endo sed na a i es, iden i ied ea u es in hem and in e ed no m
(SDMN 2)
Rep esen a i e examples o endo sed na a i es
Iden i ied ea u es
In e ed no m
(G1, Page 3) [Con ex : P e-se ice eache s discuss whe he he pupils ge he
o mula di ec ly o hey need he able o alues]
63: J:
Tha hey a e no able o ind he o mula di ec ly...
64: M:
They a e gi ing you alues, o you o do… i is ha i hey
do no gi e you alues how you ge he o mula. Le ’s
see, wha you say is ue, you can di ec ly make he o mula.
65: L:
Yes, bu …
66: M:
Bu also you can pu he able, you do no ha e o pass o
he ex o he o mula di ec ly.
67: L:
I ell you one hing I hink, be awa e ha hey a e pupils
who need he able.
68: S:
The logical hing is o make he able i s …
69: M:
I is he mos logical …
The pupils need he
able o alues o ge
he o mula
“To each he
mos common o
necessa y mode
o
ep esen a ion
is able o
alues” (SDMN
2)
(G6, Pages 165-167) [Con ex : P e-se ice eache s discuss whe he he pupils
d aw he pa able di ec ly o hey need he able o alues]
3413: F:
And now in middle would be, h ough a e bal si ua ion
equi es nume ic alues. You ake he example you ha e pu ,
(x)=x2, hence passes o g aphical mode. This p oblem
[ e e ing o he p o ided a icles] is done wi hou ables
o alues.
3414: P:
Means ha ables can also be …
3415: E:
[add essing o he uni e si y eache as he ep esen a i e o
he g oup] We… i is ha … he p oblem 3 o example, go
you o he 3. Whe he i is ha clea , i we pu hus also we
can say ha , hey gi e you ha algeb aic bu you will
ha e o make o able. Tha he pupil does no ake 3x2
and d aws i .
[…]
3443: R:
Bu i is ha , h ough a able… Bu nobody says you o
do a able, i says you o d aw i . Ano he hing is ha
you do ha ep esen a ion h ough a able
The pupils need he
able o alues o
d aw he pa able
INT ELECT J MATH ED
h p://www.iejme.com 461
The o he (SDMN 3: “To in oduce a ma hema ical con en he eache should always ollow an es ablished
sequence”) was in e ed in wo g oups (G1 and G4), on he basis o endo sed na a i es such as hose shown
in Table 5.
Wi h espec o he SDMN 3, a p e ious con en al eady es ablished should be augh h ough a “co ec ”
sequence. A eache mus ollow ha sequence o ansmi he con en , being a “pe son who ansmi s
ma hema ical knowledge”; his knowledge is pe ec ly s uc u ed bo h in i s way o eaching and lea ning.
Bo h no ms (SDMN1 and SDMN3) a e ela ed o eache ’s ole. We can obse e in hem a dis inc ion
be ween no ms such as SDMN1, which exis in he con ex o p e-se ice eache s as lea ne s ( ela ed o hese
p e-se ice eache s’ p ac ices when hey a e he lea ne s), and no ms such as SDMN3, which exis in he
con ex o p e-se ice eache s as u u e eache s.
Finally, wi h espec o he ela ionship be ween SDM no ms and di e en eache ’s pe spec i e, ou
esul s ha e shown: on he one hand, a iew o he ma hema ics and i s lea ning ha gi e impo ance o he
able o alues as a mode o ep esen a ions commonly used in he class ooms (SDMN2), and does no gi e oo
much impo ance o communica ion (SDMN5). On he o he hand, o ansmi , o alida e and o sol e a e
e bs ha can desc ibe ac ions, o condi ions ha cha ac e ize a eache ’s ole o hese p e-se ice eache s
(we do no y o gene alize o o he p e-se ice eache s) (SDMN1), and an emphasis on ollowing a p e-
Table 3. Rep esen a i e example o all he endo sed na a i es, iden i ied ea u es in hem and in e ed no m
(SDMN 4)
Rep esen a i e examples o endo sed na a i es
Iden i ied
ea u es
In e ed no m
(G4, Page 109) [Con ex : P e-se ice eache s discuss wha ep esen a ion modes
a e needed o sol e he p oblem]
2683: M:
Whe he o unde s and wha is he maximum heigh ,
use he pa able, how long is he ligh ? Well, you
ha e o in e ac wi h he si ua ion because ha
is no wo h
us nega i e alues because ha would mean
ha he plane has c ashed…
2684: D:
Exac ly
[…]
2689: D:
“h” nega i e would mean ha he plane would
ly below g ound le el, which is no c edible.
Then, how long is he ligh ?, he e we eally begin
wi h a non- e bal ex ual ep esen a ion, bu we
con inue wi h a g aphical ep esen a ion o
unde s and wha is …
2690: J:
The alues allowed o he unc ion …
2691: D:
Exac ly, because eally wi h he g aphical
exp ession, wi h he ma hema ical exp ession, i
is no su icien because he ma hema ical
exp ession allows any alue. Howe e , he
g aphic exp ession, he g aph i sel
would no make sense o example d aws i
below g ound
2692: M:
So you ha e o in e ac wi h he e bal si ua ion
We ha e o
in e ac wi h he
si ua ion o know
he ma hema ical
esul co ec
“A ma hema ical
esul is (o is no )
co ec depending
on si ua ion”
(SDMN 4)
Table 4. Rep esen a i e example o all he endo sed na a i es, iden i ied ea u es in hem and in e ed no m
(
SDMN 5)
Rep esen a i e example o endo sed na a i e
Iden i ied ea u es
In e ed no m
(G7, Page 58) [Con ex : P e-se ice eache s discuss whe he hey ha e o explain
he answe s o he ask o no ]
1474: D:
Come you o ano he [p oblem] …
1475: A:
Bu we will ha e o explain, no ?
1476: D:
You a e going o lose all a e noon.
1477: A:
Bu whe he his is silly. Whe he ha is done in a
second… wha we will do.
1478: M:
Wha we will do is nonsense.
1479: A:
Exac ly, whe he i is be e o discuss his [ e e ing o
o he p oblem], igh ?
I is no necessa y o
explain he answe s
o he ask, because i
is a was e o ime
“Explana ions
in he answe s
o he asks a e
no necessa y
because ime is
was ed”
(SDMN
5)
Toscano e al.
462 h p://www.iejme.com
es ablished sequence on he p esen a ion o ma hema ical school con en (SDMN3). In some way, we could
say ha all hese aspec s a e ela ed o some cha ac e is ics iden i ied in a eache ’s adi ional pe spec i e.
Fu he mo e, he SDMN 4 leads us in e ea u es ela ed o pe cep ion-based pe spec i e. The ma hema ical
backg ound o hese eache s can p omo e he use o ela ionships be ween he di e en modes o
ep esen a ions o gi e sense o he si ua ion posed.
DISCUSSION AND FURTHER CONSIDERATIONS
Ou esul s ex end he wo k o au ho s who ha e deal wi h di e en ypes o no ms inco po a ing he
de ini ion, iden i ica ion and s udy o socio-didac ic-ma hema ical no ms. Following au ho s such as Ta sis
and Koleza (2008) o Yackel (2001), among o he s, his esea ch deals wi h no ms ha a e no es ablished
be ween s uden s and he eache in he class oom, bu ha a ise in he in e ac ion be ween he s uden s
hemsel es (p e-se ice eache s in his s udy) when a ask is p oposed, in o de o minimize he eache ’s
in luence on he discou se gene a ed in he esolu ion p ocess o he ask (Ta sis, 2007).
In ela ion o he wo ks in which he pa icipan s we e p e-se ice eache s (among o he s, Dixon,
And easen, & S ephan, 2009; McNeal & Simon, 2000; Roy, Tobias, Sa i, & Dixon 2014; Ta sis & Koleza, 2008;
Sánchez & Ga cía, 2014), a con ibu ion o his s udy is he inco po a ion o didac ic-ma hema ical asks as a
new elemen ha allows o si ua e hese p e-se ice eache s in hei u u e p o essional ac i i y. Ou s udy
has enabled us o in o m esea che s and eache educa o s abou wha eally happens in he in e ac ion
gene a ed among he p e-se ice eache s when hey y o sol e his kind o asks. The p ocess o sol ing
hese asks has allowed us o access u u e eache s’ discou se ela ed o ma hema ics, i s lea ning and
eaching, and he no ms ha egula e i .
F om his discou se, we ha e been able o iden i y SDM no ms. Some o hem could be closely ela ed o
Spanish educa ional con ex ha emphasizes he use o some ep esen a ion modes o e o he s and o e
ansla ions be ween modes; in addi ion, i does no p omo e ma hema ical communica ion as an impo an
elemen in s uden s’ ma hema ical educa ion. O he SDMNs a e ela ed o hei way o conside ing eache ’s
ole, p o iding in o ma ion abou cha ac e is ics ha hese u u e eache s associa e wi h his ole.
The iden i ica ion o hese no ms can p o ide in o ma ion o esea che s and eache educa o s in a double
sense. On he one hand, showing how ce ain no ms, which can come om p e ious school con ex s o hese
p e-se ice eache s, can in luence hei u u e wo k. On he o he hand, highligh ing how ce ain no ms may
appea in e y di e en school con ex s, showing a common backg ound ha may ha e a sociohis o ical o igin
Table 5. Rep esen a i e example o all he endo sed na a i es, iden i ied ea u es in hem and in e ed no m
(SDMN 3)
Examples o endo sed na a i es
Iden i ied ea u es
In e ed no m
(G4, Page 114) [Con ex : P e-se ice eache s discuss he o de in which o p esen
he p oblem o hei u u e pupils]
2784: D:
Look, I p opose he ollowing, i s a able… a e wa ds
he ep esen a ion o one o he a iables ha we
can call heigh wi h espec o he o he one which
is boiling empe a u e...
2785: M:
Okay.
2786: D:
Then we al eady ha e he e ha we can p esen … o
he pupils… i s he da a p ocessing o he able
and hen a g aphical ep esen a ion o he p oblem ...
2787: D:
Okay.
P e-se ice eache s
iden i y wha appea s
o hem o be a
co ec sequence ha
is assumed wi hou
posing mo e
al e na i es
“To in oduce a
ma hema ical
con en he
eache should
always ollow
an es ablished
sequence”
(SDMN 3)
(G4, Page 116) [Con ex : P e-se ice eache s discuss he o de in which o p esen
he p oblem o hei u u e pupils]
2845: J:
Fi s ma hema ical elemen he ep esen a ion in
able o alues, and i s cons uc ion by means o a
speci ica ion, you know… he p ocess o … We will ake
ad an age o ha e e y housand down, hen he
second ma hema ical elemen is he g aphical
ep esen a ion o he linea unc ion as a ela i e
line in he plane. The ollowing is, ule o
p opo ionali y ... Do you ag ee?
2846: D:
Yes
P e-se ice eache s
iden i y wha appea s
o hem o be a
co ec sequence ha
is assumed wi hou
posing mo e
al e na i es
INT ELECT J MATH ED
h p://www.iejme.com 463
(Ma in, 2000) linked o a pa icula wes e n cul u e and sha ed by hese p e-se ice eache s. I we conside
ha we ha e iden i ied some o hese no ms in di e en g oups o p e-se ice eache s, an impo an aspec
a ises om his esul . The SDMNs migh be conside ed as some hing sha ed. This could indica e a cul u al
ea u e o he socie y in which u u e eache s ind hemsel es, ea u e ha could be ela ed o he social
conside a ion o eache s’ wo k.
Fu he mo e, we ha e iden i ied ea u es o de e mina e eache s’ pe spec i es h ough hese no ms. The
ela ionship be ween no ms and pe spec i es could be e lec ed in p e-se ice eache s’ u u e p o essional
wo k. We need o con inue esea ch on school no ms a any le el o imp o e he o e all ision o wha ole
no ms ha e in he eaching/lea ning p ocess. In pa icula , u he esea ch is needed o de e mine i hese
pe spec i es a e ans e ed o no o hei p ac ice as u u e eache s (Roy e al., 2014).
O he ques ions a e ela ed o he ask. We wonde whe he di e en asks migh gi e ise o di e en
SDMNs ha a e ela ed o o he cha ac e is ics o he pe spec i es. Fu he mo e, in ou case, he lack o
cohe ence be ween p e ious asks ha hese p e-se ice eache s ha e expe ienced in hei educa ional pa h
and he ac o ha ing o sol e a ask si ua ed in a p o essional ac i i y can a o he eme gence o he no ms
iden i ied he e.
To sum up, he sociocul u al app oach adop ed in his s udy has con ibu ed o p o ide explana ion abou
he esul s ha ha e been ob ained unde o he heo e ical app oaches. In pa icula , how SDM no ms
con ibu e o he gene a ion o eache s’ pe spec i es. I we ake in o accoun ha u u e seconda y
ma hema ics eache s a e key elemen s in he imp o emen o ma hema ics educa ion, no ms and
pe spec i es become ele an elemen s ha should be aken in o conside a ion in he eaching/lea ning
p ocesses. The implemen a ion o new app oaches and me hodologies in ma hema ics eache educa ion
p og ams can be minimized by hese no ms and pe spec i es.
We wan o highligh ing ha his wo k has no in ended in any case o gene alize, bu app oaching he
eali y o a ew p e-se ice eache s in a speci ic con ex , in o de o iden i y some aspec s o u he s udies.
Addi ional In o ma ion
An ea lie e sion o his pape was p esen ed as a Resea ch Repo a he 37 h Con e ence o he
In e na ional G oup o he Psychology o Ma hema ics Educa ion (PME).
Disclosu e s a emen
No po en ial con lic o in e es was epo ed by he au ho s.
No es on con ibu o s
Rocío Toscano – Depa amen o de Didác ica de las Ma emá icas, Uni e sidad de Se illa, Spain.
Vic o ia Sánchez – Depa amen o de Didác ica de las Ma emá icas, Uni e sidad de Se illa, Spain.
Me cedes Ga cía – Depa amen o de Didác ica de las Ma emá icas, Uni e sidad de Se illa, Spain.
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