Attitudes towards mathematics at secondary level: development and structural validation of the scale for assessing attitudes towards mathematics in secondary education (SATMAS)

Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 - 557 - h p://dx.doi.o g/10.14204/ej ep.40.15163 A i udes owa ds ma hema ics a seconda y le el: De elopmen and s uc u al alida ion o he Scale o Assessing A i udes owa ds Ma hema ics in Seconda y Educa ion (SAT- MAS) La a Yáñez-Ma quina1, Lou des Villa dón-Gallego 1 1 Facul y o Psychology and Educa ion, Uni e si y o Deus o, Bilbao Spain Co espondence: La a Yáñez-Ma quina. Facul y o Psychology and Educa ion, Uni e si y o Deus o. A da. de las Uni e sidades, 24. 48007 Bilbao (Spain). E-mail: la a.yannez@deus o.es © Educa ion & Psychology I+D+i and Ilus e Colegio O icial de la Psicología de Andaluacía O ien al (Spain) La a Yáñez-Ma quina & Lou des Villa dón-Gallego - 558 - Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 h p://dx.doi.o g/10.14204/ej ep.40.15163 Abs ac In oduc ion. In seconda y educa ion, s uden s’ low achie emen and engagemen in ma he- ma ics a e closely ela ed o hei a i udes owa ds he subjec . Despi e he in e na ional body o esea ch, an exhaus i e li e a u e e iew o he exis ing ins umen s o measu ing i d aws a en ion o he inconsis ency in he de ini ion and co esponding ac o s uc u e o he con- s uc a i udes owa ds ma hema ics. The e o e, he aim o his pape is o de elop and ali- da e an ins umen o measu ing seconda y s uden s’ a i udes owa ds ma hema ics based on a p elimina y de ailed heo e ical amewo k. Me hod. The sample comp ised 792 s uden s, wi h an a e age age o 13.96 (SD = 1.09) yea s, om Biscay (Basque Coun y Au onomous Region, Spain). Con i ma o y ac o anal- yses we e conduc ed o es he heo e ical p oposed non-hie a chical s uc u e, consis ing o h ee i s -o de ac o s (s uden ’s ma h sel -concep , pe cei ed use ulness o ma hema ics and in e es o ma hema ics). Resul s. The esul s la gely con i med ha his model showed a good i o he da a. In e nal consis ency, disc iminan alidi y and c i e ion- ela ed alidi y es s yielded good C onbach’s alpha coe icien s, s ong co ela ions be ween he p oposed dimensions and mode a ely posi- i e co ela ion sco es be ween a i udes owa ds ma hema ics and s uden s’ ma hema ical pe o mance measu ed wi h a ma h achie emen es de eloped ad-hoc. Discussion. The esul ing 19-i em scale may ep esen a psychome ically sound ins umen bo h o esea ch pu poses and o educa ional in e en ions. To conclude, he con ibu ion o he p esen s udy o he esea ch on a i udes owa ds ma hema ics is discussed, and some is- sues a e sugges ed o u u e esea ch o add ess. Keywo ds: A i udes owa ds ma hema ics, seconda y educa ion, ma hema ics educa ion, con i ma o y ac o analysis. Recep ion: 11.28.15 Ini ial accep ance: 12.14.15 Final accep ance: 10.16.16 A i udes owa ds ma hema ics a seconda y le el: De elopmen and s uc u al alida ion o he Scale o Assessing A i udes owa ds Ma hema ics in Seconda y Educa ion (SATMAS) Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 - 559 - h p://dx.doi.o g/10.14204/ej ep.40.15163 Resumen In oducción. El bajo log o y el escaso comp omiso en ma emá icas de los es udian es de secunda ia es án es echamen e elacionados con sus ac i udes hacia la asigna u a. A pesa de la amplia in es igación exis en e al espec o, una e isión exhaus i a de la li e a u a sob e ins umen os pa a su medición concluye con la inconsis encia en la de inición de la concep- ualización eó ica y co espondien e es uc u a ac o ial del cons uc o ac i udes hacia las ma emá icas. Po an o, el obje i o de es e es udio es desa olla y alida un ins umen o pa a la medición de las ac i udes hacia las ma emá icas de es udian es de secunda ia basado en un de allado ma co eó ico p e io. Mé odo. La mues a es u o compues a po 792 es udian es, con una edad media de 13.96 (DE = 1.09) años, p oceden es de Bizkaia (Comunidad Au ónoma del País Vasco, España). Se lle a on a cabo análisis ac o iales con i ma o ios pa a es a la es uc u a eó ica no je a qui- zada p opues a, consis en e en es ac o es elacionados de p ime o den (au oconcep o ma- emá ico del es udian e, u ilidad pe cibida de las ma emá icas e in e és hacia las ma emá icas). Resul s. Los esul ados con i ma on que es e modelo p esen aba buenos índices de bondad de ajus e. Los análisis co espondien es a consis encia in e na, alidez disc iminan e y alidez de c i e io a oja on buenos coe icien es al a de C onbach, co elaciones signi ica i as en e las dimensiones p opues as y co elaciones mode adas y posi i as en e las ac i udes hacia las ma emá icas y el endimien o ma emá ico de los es udian es, que ue medido con un es ma- emá ico desa ollado ad-hoc. Discussion. La escala esul an e, de 19 í ems, ep esen a un ins umen o psicomé icamen e obus o an o pa a ines de in es igación como pa a in e enciones educa i as. Pa a conclui , se discu e la con ibución del p esen e es udio al ámbi o de las ac i udes hacia las ma emá i- cas, y se sugie en algunos aspec os pa a u u as líneas de in es igación. Palab as Cla e: Ac i udes hacia las ma emá icas, educación secunda ia, educación ma emá- ica, análisis ac o ial con i ma o io. Recibido: 28.11.15 Acep ación Inicial: 14.12.15 Acep ación inal: 16.10.16 La a Yáñez-Ma quina & Lou des Villa dón-Gallego - 560 - Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 h p://dx.doi.o g/10.14204/ej ep.40.15163 In oduc ion S uden s’ ma hema ical unde pe o mance has become wo isome in many coun ies (Lipne ich, MacCann, K umm, Bu us, & Robe s, 2011). Despi e he g owing impo ance o ma hema ical hinking and ma hema ics- ela ed skills o an indi idual’s ull de elopmen in oday’s socie y, his subjec is pe cei ed by mos s uden s as abs ac , di icul , bo ing and wi hou ela ion o asks o e e yday li e (Ignacio, Nie o, & Ba ona, 2006). In such a con ex , socio-cogni i e heo ies ha e sugges ed ha s uden s’ belie s and expec a ions a e a majo de e minan o hei pu sui o and engagemen in ma hema ics cou ses (C ombie e al., 2005; G oo enboe & Hemmings, 2007; Malmi ou i, 2007). Howe e , hose a i udes a e no inna e bu o med o e ime by expe iences, declining much o e he ansi ion om uppe elemen- a y school o junio high school (Wa , 2000). An in e na ional body o esea ch has highligh ed he close ela ionship be ween s u- den s’ d opou a es in ma hema ics and bo h hei p esen and u u e ma hema ical pe o - mance (Bouchey & Ha e , 2005; Anjum, 2006; Skaal ik & Skaal ik, 2006; Samuelsson & G ans om, 2007; Kadije ich, 2008; Williams & Williams, 2010; Lipne ich, MacCann, K umm, Bu us, & Robe s, 2011). In his line, pa icula ly in e es ing is he me a-analysis ca ied ou by Ma and Kisho (1997) wi h longi udinal modelling. The indings sugges ed ha a i udes owa ds ma hema ics exe ed causal e ec s on ma hema ics achie emen . Howe e , he co esponding obse ed e ec sizes we e ound o be small, which was explained by he au ho s as a consequence o ce ain psychome ic limi a ions in he ins umen s designed o measu e a i udes owa ds ma hema ics. These limi s ha e been also acknowledged in mo e ecen esea ch by Zan, B own, E ans and Hannula (2006) and Lim and Chapman (2013), which asse ed ha he ac o s uc u e o a i udes owa ds ma hema ics emains ambiguous. As seen in Table 1, a deep look on he psychological li e a u e p o ides a numbe o di e ing concep ualiza ions o he cons uc a i udes owa ds ma hema ics, which has esul ed in many ins umen s a ge ing o measu e i . These measu emen s o assessing a i udes owa ds ma hema ics ha e been d awn om pee - e iewed a icles. Measu emen s o assessing a i- udes owa ds science o s a is ics ha e no been conside ed in his li e a u e e iew because ecen e idence sugges s ha a i udes owa ds hese h ee subjec s show di e en ajec o ies o e adolescence, meaning ha each cons uc should be in es iga ed sepa a ely (Ba h e al., 2011). A i udes owa ds ma hema ics a seconda y le el: De elopmen and s uc u al alida ion o he Scale o Assessing A i udes owa ds Ma hema ics in Seconda y Educa ion (SATMAS) Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 - 561 - h p://dx.doi.o g/10.14204/ej ep.40.15163 F om he exis ing ins umen s, he Fennema-She man Ma hema ics A i ude Scales (FSMAS; Fennema & She man, 1976) ha e been he mos widely used ac oss all le els o he ma hema ics cu iculum. Since i s de elopmen , his se o nine subscales has been ansla ed in o se e al languages o i s use wi h samples om di e en sociodemog aphic backg ounds. Ne e heless, O’Neal, E nes , McLean, and Temple on (1988) ha e yielded poo alidi y and eliabili y sco es, concluding ha he o iginal subscales migh no p ope ly gauge he esea ch cons uc . In line wi h his, Melancon, Thompson, and Becnel (1994) we e unable o ind a sui able model i o he o iginal s uc u al p oposal by Fennema and She man (1976) and yielded a mo e pa simonious s uc u e, consis ing o eigh ac o s. Likewise, Mulhe n and Race (1998) p oposed a sho ened e sion o six sepa a e ac o s, which yielded be e in e - nal consis ency on bo h he ull scale and unde lying subscales. Ano he in e es ing ins umen is he A i udes Towa d Ma hema ics In en o y (ATMI; Tapia & Ma sh, 2004), which has a mo e dis inc and cohesi e ac o s uc u e, assessed by bo h explo a o y and con i ma o y analyses. Howe e , his 40-i em scale is oo ime demand- ing. The e o e, in o de o educe he ime equi ed o i s adminis a ion, Lim and Chapman (2013) de eloped a sho ened e sion. The con i ma o y analyses yielded sound p ope ies, bu a high co ela ion coe icien was ound be ween he enjoymen and mo i a ion dimen- sions ( = .96). This esul indica ed ha hese wo la en ac o s we e s a is ically isomo phic and he e o e, a educ ion o he ac o s uc u e o h ee ac o s would p esumably yield a be e model i o he da a. On he o he hand, some ins umen s, al hough p ima ily de eloped o measu e a i udes owa ds ma hema ics, ac ually comp ise in he same scale bo h a i udinal dimensions (e.g., mo i a ion, pe cei ed use ulness) and ma hema ics anxie y. Tha is he case o he Fennema- She man Ma hema ics A i ude Scales (FSMAS; Fennema & She man, 1976), Ma hema ics A i ude In en o y (MAI; Sandman, 1980), Escala de Ac i udes hacia las Ma emá icas (EAM; Auzmendi, 1992), Escala de Ac i udes hacia la Ma emá ica-Uni e sidad (EAHM-U; Bazán, 1997) and Sho o m o Ma hema ics A i ude Scale (Yasa , 2014). Ne e heless, as E ans (2006) claimed, a i udes owa ds ma hema ics and ma hema ics anxie y a e wo sepa a e sub- domains o he mo e gene al domain ma hema ical a ec . This means ha he cons uc a i- udes owa ds ma hema ics has i s own ac o s uc u e and i s assessmen s should be es ed sepa a ely om ma hema ics anxie y. La a Yáñez-Ma quina & Lou des Villa dón-Gallego - 562 - Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 h p://dx.doi.o g/10.14204/ej ep.40.15163 Mo eo e , esea ch on he subdomain a i udes owa ds ma hema ics has ex ensi ely acknowledged i s mul idimensional na u e and has iden i ied s uden ’s con idence as a salien unde lying a iable (Ru ell, Mason, & Allen, 1998; Gómez-Chacón, 2000; Hanulla, 2002; Di Ma ino & Zan, 2010). In some cases, s uden ’s con idence is no included, such as in he Ma h A i ude Scale (Aiken & D ege , 1961), he Du on Scale (DAS; Du on & Blum, 1968), Enjoymen and Value scales (E and V scales; Aiken, 1974), Ins umen measu ing ce ain a i udes owa d ma hema ics (Michaels & Fo sy h, 1977) and Cues iona io pa a medi las ac i udes hacia las ma emá icas en alumnos de ESO (Muñoz & Ma o, 2006). In o he cases, his a iable appea s di ided in o wo sub ac o s, such as in he scale o measu ing a i udes owa d ma hema ics in Compulso y Seconda y Educa ion (Alemany-A ebola & La a, 2010) and Escala de Ac i udes hacia las Ma emá icas (EAM; Palacios, A ias, & A ias, 2014). In he o me , he au ho s dis inguished be ween posi i e and nega i e sel -concep ; in he la e , he au ho s dis inguished be ween he pe cei ed ma hema ical incompe ence and sel -concep . Howe e , due o hei na u e, in bo h cases, he wo sub ac o s conside ed by he au ho s should be cons i uen o he same homogeneous dimension. The cu en s udy D awing on he li e a u e e iew, he comp ehensi e analysis o he con en o Table 1 sugges s he i em edis ibu ion in h ee main ca ego ies: s uden ’s ma h sel -concep , pe - cei ed use ulness o ma hema ics and in e es o ma hema ics. Table 1. Measu emen s o assessing a i udes owa ds ma hema ics Ins umen Dimensions/I ems Psychome ic e i- dence Ma h A i ude Scale (Aiken & D ege , 1961) A unidimensional 20-i em scale consis ed o 10 i ems con- no ing nega i e eelings and 10 i ems conno ing posi i e eelings. EFA Tes - e es eliabil- i y: .94 (N=127) The Du on Scale (DAS; Du on & Blum, 1968) A homogeneous scale consis ing o 27 i ems ha disc imi- na e be ween posi i e and nega i e eelings abou a i hme- ic. Spea man-B own eliabili y o he ull scale: .84 (N=346) Enjoymen and Value Scales (E scale and V scale; Aiken, 1974) A se o wo scales, which can be used ei he sepa a ely o join ly: Enjoymen scale (11) and Value scale (10) EFA C onbach’s alpha o E scale: .95 C onbach’s alpha o V scale: .85 (N = 190) Fennema-She man Ma hema ics A i ude A se o nine subscales, which can be used ei he sepa a ely o join ly: Ma hema ics anxie y scale (12), a i ude owa d EFA Spli -hal eliabili y A i udes owa ds ma hema ics a seconda y le el: De elopmen and s uc u al alida ion o he Scale o Assessing A i udes owa ds Ma hema ics in Seconda y Educa ion (SATMAS) Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 - 563 - h p://dx.doi.o g/10.14204/ej ep.40.15163 Scales (FSMAS; Fenne- ma & She man, 1976) success in Ma hema ics scale (12), con idence in lea ning Ma hema ics scale (12), e ec ance mo i a ion in Ma hema - ics scale (12), a he scale (12), mo he scale (12), Ma he- ma ics as a male domain (12), eache scale (12), use ulness o Ma hema ics scale (12). ( o he subscales): .86-.93 (N=1,600) Ins umen measu ing ce ain a i udes owa d ma hema ics (Michaels & Fo sy h, 1977) A se o ou subscales: enjoymen o wo d p oblems (1), enjoymen o pic o ial p oblems (1), app ecia ion o he u ili y o ma hema ics (10), secu i y wi h ma hema ics (10), EFA Spea man-B own eliabili y ( o he subscales): .51-.78 (N=299) Ma hema ics A i ude In en o y (MAI; Sand- man, 1980) A se o six subscales, esul ing in a o al o 48 i ems: pe - cep ion o Ma hema ics eache s, alue o Ma hema ics, sel -concep in Ma hema ics, ma h anxie y, enjoymen o Ma hema ics, mo i a ion in Ma hema ics. EFA Reliabili y ( o he subscales): .69-.89 (N=5,034) Escala de ac i ud de ca ác e e bal (Gai ín, 1990) A 22-i em scale wi h h ee unde lying ac o s: liking, use- ulness, and con idence-anxie y. EFA Tes - e es eliabili y ( o he dimensions): .77-.93 (N=3,637) Escala de Ac i udes hacia las Ma emá icas (EAM; Auzmendi, 1992) A 25-i em scale wi h i e unde lying ac o s: use ulness (5), con idence (5), anxie y (5), liking (5), mo i a ion (5). EFA (PCA and VR) C onbach’s alpha ( o he dimensions): .50-.91 C onbach’s alpha o he ull scale: .93 (N=1,221) Escala de Ac i udes hacia la Ma emá ica- Uni e sidad (EAHM-U; Bazán, 1997) A 31-i em scale wi h ou unde lying ac o s: a ec i i y (8), applicabili y (8), abili y (8) and anxie y (7). EFA C onbach’s alpha ( o he dimensions): .71-.91 A i udes owa d Ma he- ma ics and Ma hema ics Taugh wi h Compu e (AMMEC; U sini, Sánchez, & O endain, 2004) A 29-i em scale comp ising h ee subscales: liking o ma h- ema ics (11), liking o ma hema ics augh wi h compu e (11), sel -con idence (7). EFA (PCA and VR) C onbach’s alpha ( o he subscales): .68-.81 C onbach’s alpha o he ull scale: .80 Spli -hal eliabili y o he ull scale: .71 (N=439) A i udes owa d Ma he- ma ics In en o y (ATMI; Tapia & Ma sh, 2004) A 40-i em scale wi h ou unde lying dimensions: sel - con idence (15), alue (10), enjoymen (10), mo i a ion (5) EFA (ML and VR) and CFA C onbach’s alpha ( o he dimensions): .88-.95 C onbach’s alpha o he ull scale: .97 Tes - e es eliabili y ( o he dimensions): .70-.80 Tes - e es eliabili y o he ull scale: .89 (N=545) Cues iona io pa a medi las ac i udes hacia las ma emá icas en alumnos A 19-i em scale wi h wo unde lying ac o s: liking and use ulness (9), eache ’s a i ude owa d ma hema ics pe - cei ed by he s uden (10) EFA (PCA and VR) C onbach’s alpha o he ull scale: .97 La a Yáñez-Ma quina & Lou des Villa dón-Gallego - 564 - Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 h p://dx.doi.o g/10.14204/ej ep.40.15163 de ESO (Muñoz & Ma o, 2006) (N=1,220) Scale o measu ing a i udes owa d ma he- ma ics in compulso y seconda y educa ion (Alemany-A ebola & La a, 2010) A 35-i em scale wi h se en unde lying ac o s: beha iou al componen (13), a ec i e componen (7), nega i e sel - concep (5), posi i e sel -concep (3), cogni i e componen (3), demo i a ion owa ds ma hema ics (2), expec ancy (2) EFA and CFA C onbach’s alpha ( o he dimensions): .43-.89 C onbach’s alpha o he ull scale: .92 (N=236) Sho ened e sion o he A i udes owa d Ma he- ma ics In en o y (sho ATMI; Lim & Chapman, 2013) A 19-i em sho ened ATMI e sion wi h ou subscales: enjoymen (5), mo i a ion (4), sel -con idence (5), pe - cei ed alue (5). EFA and CFA C onbach’s alpha ( o he subscales): .85-.90 C onbach’s alpha o he ull scale:.93 Tes - e es eliabili y o he ull scale: .75 (N=1,601) Sho o m o Ma hema - ics A i ude Scale (Yasa , 2014) A 19-i em scale wi h ou unde lying dimensions: enjoy- men (6), ea , anxie y and bo edom (5), place o ma hema - ics in li e (4), pe cei ed ma hema ics success (4). EFA (BCA) and CFA C onbach’s alpha ( o he dimensions): .82-.89 C onbach’s alpha o he ull scale: .96 (N=1,801) Escala de Ac i udes hacia las Ma emá icas (EAM; Palacios, A ias, & A ias, 2014) A 32-i em scale unde lying ou dimensions: pe cep ion o ma hema ical incompe ence (12), liking (12), pe cep ion o use ulness (4), ma hema ical sel -concep (4) EFA (PAF, ML and PR) and CFA C onbach’s alpha ( o he dimen- sions):.68-.93 C onbach’s alpha o he ull scale: .95 (N=4,807) No e. BCA=Basic Componen Analysis, CFA=Con i ma o y Fac o Analysis, EFA=Explo a o y Fac o Analy- sis, ML=Maximum Likelihood, PCA=P incipal Componen Analysis, PR=P omax Ro a ion, VR=Va imax Ro a- ion. Using his ca ego iza ion as a basis o he p esen s udy, he heo e ical model p oposed o be es ed is i-dimensional and non-hie a chized, wi h he ollowing de ini ions o he h ee ela ed i s -o de la en ac o s: S uden ’s ma h sel -concep : encompasses a b oad ange o s uden ’s esponses abou he o his abili y o lea n and do ma hema ics (e.g., “I am unable o sol e ma h p oblems”). A s uden who sco es high in his dimension belie es ha she o he has he abili y o unde s and and sol e ma h- ela ed asks. On he o he hand, a s uden sco ing low in his dimension does no belie e ha she o he has he abili y o unde s and and do ma hema ics. A i udes owa ds ma hema ics a seconda y le el: De elopmen and s uc u al alida ion o he Scale o Assessing A i udes owa ds Ma hema ics in Seconda y Educa ion (SATMAS) Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 - 565 - h p://dx.doi.o g/10.14204/ej ep.40.15163 Pe cei ed use ulness o ma hema ics: measu es he s uden s’ ex insic u ili y alue o ma hema ics. This dimension, as de ined by Eccles and Wig ield (2002), measu es s uden s’ belie s abou he applicabili y o ma hema ics o hei cu en and u u e goals and in ela ion o school, ca ee and e e yday li e (e.g., “Lea ning ma h will inc ease my u u e job oppo u- ni ies”). The e o e, a s uden who sco es high in his dimension inds ma hema ics e y use ul o bo h hei cu en and u u e goals; whe eas a s uden sco ing low inds i useless o bo h hei cu en and u u e goals. In e es o ma hema ics: e e s o he amoun o in e es s uden s ha e in lea ning and doing ma hema ics (e.g., “Time jus lies by when I am sol ing ma h p oblems”). A s uden who sco es high in his dimension has high in e es in lea ning and doing ma hema ics. On he o he hand, a s uden sco ing low dislikes ma hema ics, inds i bo ing and does no ake pleasu e om lea ning and doing ma hema ics. Me hod Pa icipan s The sample consis ed o 792 seconda y s uden s, which we e selec ed ia a clus e - sampling me hod om 36 classes om he p o ince o Biscay (Basque Coun y Au onomous Region, Spain). This sample was hen di ided in o wo subg oups acco ding o he language in which hey lea n ma hema ics (see Table 2). Table 2. Sociodemog aphic p o ile o he sample 2nd g ade 4 h g ade To al Females Males Females Males Subsample 1 45 72 88 138 343 Subsample 2 137 138 88 86 449 To al 182 210 176 224 792 Ins umen s Scale o assessing a i udes owa ds ma hema ics in seconda y educa ion (SATMAS) A pool o i ems was collec ed om all he examined ins umen s (see Table 1) and e- dis ibu ed in he a o emen ioned h ee dimensions. A e emo ing edundan i ems, ewo d- ing some o he s and adding a ew newly w i en ones, a inal pool o 36 i ems was ob ained La a Yáñez-Ma quina & Lou des Villa dón-Gallego - 572 - Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 h p://dx.doi.o g/10.14204/ej ep.40.15163 he bes s uc u e o explain he cons uc a i udes owa d ma hema ics, as i showed a sim- ple s uc u e compa ed o ha o M1a. Table 6. Goodness-o - i indices o he e ised s uc u es Model Subsample χ 2 S-B /d NNFI CFI IFI SMSR RMSEA (90% CI) AIC M1 S1 3.56 .90 .91 .91 .071 .068 [.062, .072] 354.52 S2 2.06 .90 .91 .91 .079 .068 [.059, .077] 13.683 M1a S1 2.78 .94 .95 .95 .053 .056 [.050, .062] 128.803 S2 1.62 .95 .96 .96 .058 .052 [.040, .063] -63.335 M1b S1 3.28 .93 .94 .94 .056 .064 [.057, .070] 190.994 S2 1.77 .94 .95 .95 .060 .058 [.046, .069] -34.504 M1c S1 2.88 .94 .95 .95 .053 .058 [.051, .064] 130.788 S2 1.54 .96 .96 .96 .057 .049 [.036, .061] -68.206 Final model In addi ion, all s anda dized ac o loadings and in e - ac o co ela ions we e s a is ical- ly signi ican (p < .05), wi h alues anging om .49 o .92 o ac o loadings and om .21 o .73 o in e - ac o co ela ions (see Figu e 1). These esul s unde sco ed he disc iminan alidi y o he scale be ween he h ee unde lying la en ac o s. Figu e 1. Con i ma o y ac o analysis o SATMAS wi h subsample 1 (N = 563) and subsam- ple 2 (N = 229) A i udes owa ds ma hema ics a seconda y le el: De elopmen and s uc u al alida ion o he Scale o Assessing A i udes owa ds Ma hema ics in Seconda y Educa ion (SATMAS) Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 - 573 - h p://dx.doi.o g/10.14204/ej ep.40.15163 No e. MSC=Ma h Sel -Concep , PUM=Pe cei ed Use ulness o Ma hema ics, IM=In e es o Ma hema ics. Addi ional p ope ies o he las 19-i em SATMAS we e assessed wi h he composi e eliabili y (CR) and C onbach’s coe icien ( α ) o each dimension. The eliabili y analyses showed good in e nal consis ency o he s uden ’s ma h sel -concep (CR s1 = .93, α s1 = .93; CR s2 = .89, α s2 = .89), pe cei ed use ulness o ma hema ics (CR s1 = .78, α s1 = .78; CR s2 = .80, α s2 = .80) and in e es o ma hema ics (CR s1 = .90, α s1 = .91; CR s2 = .90, α s2 = .91). Re- ga ding he ull scales, eliabili ies we e α 1 = .93 and α 2 = .91 o Subsample 1 and Subsam- ple 2, espec i ely. The eliabili y sco es we e ound o be good based on he Nunnally’s c i- e ion (Nunnally, 1978). The e o e, he esul s sugges ed ha he i ems we e in e nally con- sis en in ep esen ing he co esponding ac o . Finally, c i e ion- ela ed alidi y o he scale was assessed by he Pea son co ela ion coe icien s be ween a i udes owa ds ma hema ics and he sco es ob ained in ma hema ics achie emen es s. Posi i e signi ican co ela ions we e ound, as expec ed, be ween s u- den ’s ma h sel -concep and ma h achie emen ( s1 = .23, p < .001; s2 = .22, p < .001), pe - cei ed use ulness o ma hema ics and ma h achie emen ( s1 = .22, p < .001; s2 = .23, p < .001) and in e es o ma hema ics and ma h achie emen ( s1 = .20, p < .001; s2 = .21, p < .001). The ull scale is included in Table 7. Table 7. Las e sion o he 19-i em SATMAS S uden ’s ma h sel -concep IT01 I eel mo e oolish han my classma es when sol ing ma h p oblems and exe cises IT02 In spi e o my e o , I canno unde s and ma h IT03 I ha e di icul ies wi h ma h IT04 I was no a bo n ma h lea ne IT05 I am unable o sol e ma h p oblems IT06 Wha e e I do, I ge low g ades in ma h IT07 I will be always ha d o me o lea n ma h Pe cei ed use ulness o ma hema ics IT08 Ma h is e y use ul IT09 E e ybody needs o lea n ma h IT12 Ma h is necessa y o li e IT13 Ma h is impo an o socie y de elopmen IT15 Lea ning ma h is impo an o my u u e job In e es o ma hema ics IT17 I like s udying ma h IT18 I like ma h IT19 Time jus lies by when I am s udying ma h La a Yáñez-Ma quina & Lou des Villa dón-Gallego - 574 - Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 h p://dx.doi.o g/10.14204/ej ep.40.15163 IT20 S udying ma h is un IT21 Time jus lies by when I am sol ing ma h p oblems / exe cises IT22 Ma h is en e aining IT23 Ma h is a d ag Discussion O e he pas yea s, he e has been a g owing in e es in s udying he s uden s’ a i udes owa ds ma hema ics because o hei impo an ole in he engagemen in and mas e y o ma hema ics (e.g., McLeod, 1992; Goldin, 2002; G oo enboe & Hemmings, 2007; Malmi ou i, 2007). Ne e heless, co ela ional esea ch on a i udinal a iables and ma he- ma ical pe o mance has been con ingen on he psychome ic p ope ies o he measu emen s o measu ing he cons uc a i udes owa ds ma hema ics. Th ee conclusions may be d awn om a comp ehensi e li e a u e e iew on he exis ing ins umen s. Fi s ly, he mos widely ci ed ins umen s a e he FSMAS (Fennema & She man, 1976) and he ATMI (Tapia & Ma sh, 2004), which ha e also been ansla ed in o se e al languages o hei use in back- g ounds wi h di e en socio-cul u al cha ac e is ics. Howe e , subsequen eplica ion s udies o hese ins umen s (e.g., O’Neal, E nes , McLean, & Temple on, 1988; Melancon, Thomp- son, & Becnel, 1994; Mulhe n & Race, 1998) ha e ob ained e idence ha ebuilding some o hei la en ac o s and sho ening he scales o ewe subdomains would yield a be e i o da a. Secondly, he e a e some ins umen s (i.e., MAI, Sandman, 1980; EAM, Auzmendi, 1992; EAHM-U, Bazán, 1997; Sho Fo m o Ma hema ics A i ude Scale, Yasa , 2014) ha , al hough p ima ily se o measu e a i udes owa ds ma hema ics, ac ually mix bo h a i udinal ac o s and ma hema ics anxie y. Ne e heless, a i udes owa ds ma hema ics and ma hema - ics anxie y a e claimed o be conside ed as wo sepa a e subdomains o he mo e gene al do- main ma hema ical a ec (E ans, 2006). This means ha a i udes owa ds ma hema ics has i s own ac o s uc u e and i s assessmen s should be es ed sepa a ely om ma hema ics anx- ie y. Thi dly, hey a e o he measu emen s ha do no include s uden ’s sel -con idence as an unde lying ac o (i.e., in he Ma h A i ude Scale, Aiken & D ege , 1961; DAS, Du on & Blum, 1968; E and V Scales, Aiken, 1974; Ins umen measu ing ce ain a i udes owa d ma hema ics, Michaels & Fo sy h, 1977; and Cues iona io pa a medi las ac i udes hacia las ma emá icas en alumnos de ESO, Muñoz & Ma o, 2006) o ha include i in such a way ha emains ambiguous (i.e., in he scale o measu ing a i udes owa ds ma hema ics in Compul- A i udes owa ds ma hema ics a seconda y le el: De elopmen and s uc u al alida ion o he Scale o Assessing A i udes owa ds Ma hema ics in Seconda y Educa ion (SATMAS) Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 - 575 - h p://dx.doi.o g/10.14204/ej ep.40.15163 so y Seconda y Educa ion, Alemany-A ebola & La a, 2010; and EAM, Palacios, A ias, & A ias, 2014). As a esul o his lack o consis ency in de ining he heo e ical concep ualiza- ion o he cons uc a i udes owa ds ma hema ics, his pape aims o de elop and alida e, by means o con i ma o y echniques, a s uc u al model o his cons uc . D awing on he comp ehensi e analysis o he exis ing ins umen s, h ee la en ac o s a e es ed as i s -o de cons i uen s o he a i udes owa ds ma hema ics, conside ing his as a non-hie a chized s uc u e: a) s uden s’ ma h sel -concep , b) pe cei ed use ulness o ma he- ma ics, and c) in e es o ma hema ics. The sample, consis ing o 792 compulso y seconda y s uden s, was di ided in o wo subg oups acco ding o he language in which hey lea ned ma hema ics (Spanish o Basque). Goodness-o - i indices we e calcula ed o he heo e ical model and u he imp o emen s we e made a e examining s anda dized ac o loadings, modi ica ion co a iance indices and s anda dized esidual co a iance sco es. The las e sion, ob ained a e elimina ing ou o iginal i ems and o cing o co a ia e o ano he wo pai s o i ems, yielded a be e model i . The goodness-o - i indices we e ound o be e y simila in bo h s uden subg oups. In e nal consis ency, measu ed by C onbach’s coe icien , had an a e age alue o .92, which was e e ed o as excellen acco ding o Nunally’s c i e ion (Nunnally, 1978). Disc iminan and c i e ion- ela ed alidi ies we e also assessed. On he one hand, in e - ac o co ela ion sco es we e posi i e and anged om .21 o .73, meaning ha he la en ac o s we e no s a is ically isomo phic. On he o he hand, bi a ia e co ela ion analyses yielded signi ican posi i e co ela ion sco es be ween each a i udinal dimension and he sco e ob ained in he ma hema ical achie emen es , de eloped ad-hoc. Al hough s a is ically signi ican , hese co ela ion sco es we e ound o be small, which was explained by he e ec ha o he a iables, no conside ed in his pape , had on s uden s’ ma hema ical achie emen . In ac , a li e a u e e iew d aws a en ion o he ac ha ma hema ics achie e- men is a ec ed no only by cogni i e and a i udinal ac o s, bu also by school en i onmen (C eeme s & Reezig , 2006) o ins uc ional s a egies (Van de G i & Hou een, 2006). Despi e he p omising indings, he e a e also some me hodological limi a ions ha wa - an cau ious conside a ions in gene alizing he esul s. Fi s , da a we e collec ed solely om Biscay (Basque Coun y Au onomous Region, Spain), meaning ha he esul s a e no en i ely gene alizable ou side o a Basque popula ion. Ne e heless, he e a e some pieces o e idence sugges ing ha i is possible o ely on he quali y o he da a. On he one hand, he s uden La a Yáñez-Ma quina & Lou des Villa dón-Gallego - 576 - Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 h p://dx.doi.o g/10.14204/ej ep.40.15163 sample was selec ed ia a clus e -sampling me hod, and he desc ip i e s a is ics o he esul - ing g oup we e e y simila o ha o he e e ence popula ion, acco ding o he o icial en- olmen da a o he Educa ion Depa men o Basque Go e nmen o he school yea in which he s udy was ca ied ou . As a esul , he sample g oup was ep esen a i e and la ge enough o he esea ch pu poses, al hough no comple ely p obabilis ic in case o gene aliz- ing he indings ou side Biscay. In his line, u u e esea ch using la ge samples om di e - en sociodemog aphic con ex s would be necessa y o u he assess he in a iance o he ac- o s uc u e. On he o he hand, his s udy p o ides s ong eliabili y and alidi y e idence o he ins umen (i.e., cons uc alidi y, disc iminan alidi y, c i e ion- ela ed alidi y and in- e nal consis ency), bu i would be in e es ing o assess he s uc u al s abili y h ough es - e es eliabili y analyses. Gi en he eliabili y and alidi y e idence ga he ed in he p esen s udy, he 19-i em SATMAS p o es o be a p omising ins umen o assessing seconda y s uden s’ a i udes owa ds ma hema ics. On he one hand, he esul s la gely suppo ed he heo e ical concep u- aliza ion acco ding o which a i udes owa ds ma hema ics is a mul idimensional cons uc wi h a non-hie a chized s uc u e consis ing o he h ee a o emen ioned i s -o de ac o s (namely, s uden ’s ma h sel -concep , pe cei ed use ulness o ma hema ics and in e es o ma hema ics). This was ound o be a g ea con ibu ion o he esea ch on a i udes owa ds ma hema ics. On he o he hand, he de eloped scale is easy o adminis e and no ime- demanding, as his sho o m akes seconda y s uden s 15 minu es o comple e. The e o e, ei he school counselo s o educa o s migh use i o measu e s uden s’ a i udes owa ds ma hema ics and p o ide ea ly a en ion measu es in case he le els o ma hema ical sel - concep and mo i a ion we e low. This migh be o g ea in e es pa icula ly o hose s u- den s showing s ong ma hema ical skills bu s uggling wi h low expec a ions and in e es , which pu hem a isk o disengagemen in ma hema ics and ma hema ics- ela ed pa hways. In ac , since a i udes owa ds he subjec ha e been shown o decline om uppe elemen a y school o junio high school, de eloping and alida ing a measu emen a ge ing seconda y s uden s would u nish insigh s wi hin he ield o ma hema ics educa ion and may well be- come a s a ing poin o iden i y and p e en hose d opou isky si ua ions a ma hema ics class ooms. On o he hand, esea che s migh use i as he s a ing poin o iden i y he key domains o a i udes owa ds ma hema ics a ec ing he ma hema ical achie emen and u he in es iga e he a iables which a ec hei p e alence. In ac , assessing he dimensions un- A i udes owa ds ma hema ics a seconda y le el: De elopmen and s uc u al alida ion o he Scale o Assessing A i udes owa ds Ma hema ics in Seconda y Educa ion (SATMAS) Elec onic Jou nal o Resea ch in Educa ional Psychology, 14(3), 557-581. ISSN: 1696-2095. 2016. no. 40 - 577 - h p://dx.doi.o g/10.14204/ej ep.40.15163 de lying a i udes owa ds ma hema ics is c i ical o gain knowledge abou he plausible ac- o s ha a ec ma hema ical achie emen . Acknowledgemen s This esea ch was unded by he Uni e si y o Deus o Resea ch T aining G an s P o- g amme 2013-2016 (Uni e si y o Deus o, Bilbao). Re e ences Aiken, L.R. (1974). Two scales o a i ude owa d ma hema ics. 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