Las actitudes hacia las matemáticas: construcción y validación de un instrumento para su medida

27 Pages·2015·0.33 MB·Spanish

Revista de Psicodidáctica
ISSN: 1136-1034
[email protected]
Universidad del País Vasco/Euskal Herriko
Unibertsitatea
España
Palacios, Andrés; Arias, Víctor; Arias, Benito
Las actitudes hacia las matemáticas: construcción y validación de un instrumento para su medida
Revista de Psicodidáctica, vol. 19, núm. 1, enero-junio, 2014, pp. 67-91
Universidad del País Vasco/Euskal Herriko Unibertsitatea
Vitoria-Gazteis, España
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Revista de Psicodidáctica, 2014, 19(1), 67-91 ISSN: 1136-1034 eISSN: 2254-4372
www.ehu.es/revista-psicodidactica © UPV/EHU
DOI: 10.1387/RevPsicodidact.8961
Attitudes Towards Mathematics:
Construction and Validation of a Measurement
Instrument
Andrés Palacios*, Víctor Arias**, and Benito Arias*
University of Valladolid (Spain)*, University of Talca (Chile)**
Abstract
The measure of attitudes towards mathematics is a valuable area within the so-called affective domain
in mathematics due to the number of investigations and the extension of their relations. However, most
of the instruments currently available to measure these attitudes are validated by not overly robust
psychometric procedures and, sometimes, in not very large samples. Using a sample of 4,807 students
of all the non-university levels and following both the classical test theory, structural equation models
(measurement models), and the proposal of the item response theory (graded response model), a solid
and robust instrument to measure attitudes towards mathematics is presented, with contrasted evidence
of validity and reliability.
Keywords: Attitudes towards mathematics, exploratory factor analysis, confirmatory factor analy-
sis, graded response model, psychometrics.
Resumen
La medida de las actitudes hacia las matemáticas supone un campo de gran valor dentro de lo que se
conoce como dominio afectivo matemático por el número de investigaciones y por la amplitud de sus
relaciones. No obstante, los instrumentos disponibles en la actualidad para medir estas actitudes están
en la mayoría de los casos validados mediante procedimientos psicométricos poco robustos y, en algu-
nas ocasiones, con tamaños muestrales no muy elevados. A partir de una muestra de 4.807 alumnos de
todos los niveles no universitarios y siguiendo tanto el acercamiento de la Teoría Clásica de los Test
como los modelos de ecuaciones estructurales (modelos de medida) y el planteamiento de la Teoría de
Respuesta a los Ítems (modelo de respuesta graduada) se presenta un instrumento de medida de las acti-
tudes hacia las matemáticas sólido y robusto y con evidencias contrastadas de validez y fiabilidad.
Palabras clave: Actitudes hacia las matemáticas, análisis factorial exploratorio, análisis factorial
confirmatorio, modelo de respuesta graduada, psicometría.
Acknowledgments: This research was also supported by the Spanish Ministry of Science and Innova-
tion (EDU2009-12063).
Correspondence concerning this article should addressed to Andrés Palacios Picos. Escuela Universi-
taria de Magisterio. Plaza Alto de los Leones de Castilla. Campus María Zambrano. 40005 Segovia.
E-mail: [email protected]
68 ANDRÉS PALACIOS, VÍCTOR ARIAS, AND BENITO ARIAS
Introduction fer to the valuation, the appraisal,
and the enjoyment of this disci-
The works of McLeod (1988, pline, underlining the affective
1992) on affect in mathematics are facet more than the cognitive one.
an inflection point in the research Mathematical attitudes, in con-
in which, till this time, rational and trast, refer to the way one uses
cognitive aspects were predomi- general capacities that are relevant
nant. In one of his works (McLeod, for mathematics (such as mental
1988) which, in a sense, marks the openness, flexibility when seeking
beginning of concern with emo- solutions to a problem, reflective
tions and feelings in mathematics, thinking), aspects which are all
he established a distinction —now more closely related to cognition
classical— between attitudes, be- than to affect.
liefs, and emotions as the com- With regard to attitudes to-
ponents of what is now known as wards mathematics, their tran-
the affective domain in mathemat- scendence in the process of teach-
ics. Among these components, at- ing-learning and in students’
titudes have played a predominant mathematical performance is well
role in mathematical education due known (Miñano & Castejón, 2011;
to the number of investigations Miranda, 2012; Sakiz, Pape, &
they have generated. Hoy, 2012). The influence of pos-
Gil, Blanco, and Guerrero itive attitudes towards mathe-
(2005) note that, in the world of matics on anxiety is also well es-
mathematics, this concept of at- tablished (Akin & Kurbanoglu,
titude has been employed with 2011). In this regard, some works
a definition that is not as clear have found that students with bet-
as the one used in Psychology, ter attitudes towards mathemat-
as a predisposition with an emo- ics have higher perceptions of the
tional charge that directs and/or utility of mathematics, denoting
influences behavior; a definition intrinsic motivation towards their
that underlines three basic com- study (Perry, 2011), they have a
ponents of attitudes: cognition better mathematical self-concept
or beliefs about the target of the (Hidalgo, Maroto, & Palacios,
attitude, affect or the evaluative 2005), are more confident they
charge of such beliefs, and a be- can learn mathematics (McLeod,
havioral intention toward the at- 1992) and, especially, they dis-
titude. play approach behaviors towards
Nevertheless, with regard to mathematics (Fennema & Sher-
mathematics, we can distinguish man, 1976).
mathematical attitudes and atti- Due to their importance, at-
tudes towards mathematics. At- tempts to measure attitudes to-
titudes towards mathematics re- wards mathematics appear early
Revista de Psicodidáctica, 2014, 19(1), 67-91
ATTITUDES TOWARDS MATHEMATICS:
CONSTRUCTION AND VALIDATION OF A MEASUREMENT INSTRUMENT 69
on, and the works of Aiken (Aiken, most popular measure of attitudes
1972, 1974, 1979; Aiken & Dreger, towards mathematics of the last
1961), with the contributions of three decades. The origin of this
Dutton and Blum (1968), are pio- scale lies in the study of differ-
neer in this topic. ences between men and women
In one of the first measure- in their attitudes towards math-
ment instruments of these atti- ematics as well as their influence
tudes, Aiken and Dreger (1961) on performance. This scale has
prepared a questionnaire made up been the object of extensive stud-
of 20 items with two subscales: ies and it has been translated into
Pleasure and Fear of Mathemat- various languages, and modified
ics. As these dimensions can be for application in different situa-
considered the extreme poles of tions.
the same continuum, some authors The contribution of Tapia and
have considered it a unidimen- Marsh (2004), The Attitude toward
sional scale (Auzmendi, 1992). In Mathematics Inventory (ATMI), is
a later version, Aiken (1972) in- doubtless one of the most exten-
troduced the factor Enjoyment of sively used instruments to meas-
Mathematics. Two years later, the ure attitudes towards mathemat-
same author (Aiken, 1974) pre- ics. Its final version is made up of
sented what is no doubt one of 49 items that attempt to assess six
the most frequently used scales in aspects of these attitudes: Confi-
the measure of attitudes towards dence-Self-concept, Anxiety, and
mathematics, comprised of two Utility-Value of Mathematics, En-
subscales: the Value of Mathe- joyment of Mathematics, Motiva-
matics scale and the Enjoyment tion and Parents’ and Teachers’
of Mathematics scale. In a later Expectations.
version, Aiken (1979) increased Among the most recent con-
the number of factors to a total of tributions in English, we under-
four: Enjoyment of Mathematics, line the work of Kadijevich (2008),
Mathematical Motivation, Value- from the TIMSS-2003 report, as
Utility of Mathematics, and Fear well as those of Tahara, Ismailb,
of Mathematics. Zamanic, and Adnand (2010).
There have been numerous ad- Adelson and McCoach (2011) pre-
aptations of these scales (Aiken, pared a scale of attitudes towards
1974, 1979), coinciding with the mathematics for primary education
original reliability values and with students, which they called The
the factor structure of the four sub- Math and Me Survey and which,
scales. after the preliminary analyses, pre-
The scale of Fennema and sented two factors related to the
Sherman (1976) is, in the words perception of efficacy and enjoy-
of Tapia and Marsh (2004), the ment of mathematics.
Revista de Psicodidáctica, 2014, 19(1), 67-91
70 ANDRÉS PALACIOS, VÍCTOR ARIAS, AND BENITO ARIAS
The adaptations to Spanish of the Anglo Saxon world. Among
the scales of Aiken (1974) and of the author’s proposals is his ver-
Fennema and Sherman (1976), as bal scale made up of 22 items rated
well as the later ones of Tapia and on a Likert scale, with three di-
Marsh (2004) are scarce and gen- mensions related to enjoyment of
erally not oriented toward psycho- mathematics, utility of mathemat-
metric analysis. Such is the case ics, and confidence-anxiety to-
of the adaptation of Cazorla, Silva, wards mathematics. The reliabil-
Vendramini, and Brito (1999) of ity indexes of these three factors,
the scale of Aiken (1974) based on obtained with the test-retest tech-
a prior Portuguese scale of Brito nique, showed correlations ranging
(1998), for the study attitudes to- from .77 to .93 in the time inter-
wards statistics; the scales of val, and the reliability of the entire
Quiles (1993), which attempt to scale was .84.
relate attitudes towards mathemat- Auzmendi (1992) designed
ics and academic performance; the what is clearly the scale of atti-
more modern scale of Estrada and tudes towards mathematics that is
Díez-Palomar (2011), focused on the most cited of those created in
the mathematical education of rel- Spanish. As in Gairín (1990), the
atives; or the scale of González- author justifies the elaboration
Pienda, Fernández-Cueli, García, of a new scale on the basis of the
Suárez, Fernández, Tuero-Herrero, lack of this kind of instruments
and Helena da Silva (2012), aimed in Spanish. The final test has 25
at determining differences in the items that, after the correspond-
mathematical attitudes of men ing factor analyses, present five
and women. This lack of adapta- main components: Feelings of
tions to Spanish was noted in the Anxiety and Fear towards math-
early works of Gairín (1990) and ematics manifested by the stu-
more recently by Muñoz and Mato dent, Liking-Enjoyment of Math-
(2008, who also mentioned the in- ematics, Utility of Mathematics,
existence of adaptations of these and Motivation and Confidence.
scales in our context. Cronbach’s alpha of these scales
As mentioned, the work of ranges between .91 for the Anxi-
Gairín (1990) can be considered ety scale, and the lower value of
pioneer in the measure of attitudes .49 for the Confidence scale. The
towards mathematics in the Span- validation sample was made up of
ish language. In this work, the 1,221 secondary and high school
author mentions the need for an students.
instrument to measure attitudes to- As instruments equally dis-
wards mathematics in Spanish be- tant in terms of time, we note the
cause, at that time, all the known contributions of Escudero and
scales originally proceeded from Vallejo (1999) who prepared an
Revista de Psicodidáctica, 2014, 19(1), 67-91
ATTITUDES TOWARDS MATHEMATICS:
CONSTRUCTION AND VALIDATION OF A MEASUREMENT INSTRUMENT 71
instrument to measure attitudes to- ethnic origin and it was prepared
wards mathematics using a total both in Spanish and in the Tama-
of 18 items related to enjoyment, zight language. The final version
utility, and motivation. One year obtained a Cronbach’s alpha of .92
before, Bazán and Sotero (1998) in a sample of 236 students from
had prepared the “Escala de Ac- second and third grade of second-
titudes hacia las Matemáticas” (in ary education.
English, Attitudes towards Math- In summary, the different
ematics Scale; EAHM-V), made scales of attitudes towards math-
up of 31 items divided into four ematics, both in English and in
dimensions: Affect, Applicability, Spanish, generally present ade-
Reliability, and Anxiety. The scale quate reliability indexes if the lim-
was aimed at measuring the atti- itations of the Cronbach alpha co-
tudes of students who had just en- efficient to assess reliability are
tered the university, and obtained a not taken into account (see our
reliability of .90 for the total scale comment in the section ‘Evidence
in a sample of 256 university stu- of reliability and internal consist-
dents. ency’). Nevertheless, the disparity
In recent years, new propos- of the subscales hinders reaching
als have been made, among which a coherent—or at least a unified—
are notable the contributions of interpretation of the construct of
Muñoz and Mato (2008) and of attitudes towards mathematics.
Alemany and Lara (2010). Muñoz In a large number of these scales,
and Mato (2008) presented a scale at least in the Spanish versions,
of attitudes towards mathematics, the psychometric values were ob-
designed using a sample of 1,220 tained from small samples, and
secondary education students. The mostly from students in Compul-
final questionnaire had 19 items sory Secondary Education.
that, after the corresponding fac- In the following paragraphs,
tor analysis, presented two factors: we will present a multidimensional
Teacher’s Attitude as perceived by scale of attitudes towards math-
the student and Enjoyment-Utility ematics with items taken from the
of Mathematics. The final version corresponding references adapted
had a reliability of .97. Lastly, we to our current historical-cultural
note the contribution of Alemany setting, with a very large sample
and Lara (2010), who designed and of primary, secondary, and high
validated a new scale of attitudes school students, and with psycho-
towards mathematics for second- metric values obtained from the
ary students, made up of 37 items. proposals of the classical test the-
A differentiating element of this ory, the structural equation models
work is that the validation sample (measurement model), and the item
was made up of students of Berber response theory (Hambleton, Swa-
Revista de Psicodidáctica, 2014, 19(1), 67-91
72 ANDRÉS PALACIOS, VÍCTOR ARIAS, AND BENITO ARIAS
minathan, & Rogers, 1991; Same- data from these centers was col-
jima, 1969, 2010). lected in all the trajectories of the
selected courses. Of the partici-
pants, 67% studied in schools lo-
Method cated in province capitals, and the
remaining 34% in rural areas. The
Participants participant schools were selected
by means of stratified random sam-
The study was carried out with pling, taking the geographical area,
a sample of 4,807 students from the educational level, and the of-
14 public, private, and subsidized ficial status of the center as selec-
schools and institutes from Span- tion stratum. Participants’ mean
ish provinces. Of the 14 partici- age was 14 years, ranging from 11
pant centers, 3 were from Segovia to 23 years. Distribution by edu-
(25%), 3 from Ávila (8% of the cational levels is shown in Table
student body), 3 from Soria (10%), 1. Of the participants, 53% were
2 from Valladolid (40%), and 3 male and 47% were female. Their
from Zamora (17%). Two of these grades in mathematics had a nor-
centers were private and/or sub- mal distribution with a mean value
sidized schools or institutes. The of 5.62 (SD = 1.95).
Table 1
Participants’ Educational Level
N % % males % females
6th. Primary 394 8.20 53.8 46.2
1st. CSE 828 17.22 54.0 46.0
2nd. CSE 1,035 21.53 55.1 44.9
3rd. CSE 1,267 26.36 52.2 47.8
Valid 4th. CSE 680 14.15 49.5 50.5
1st. HE 348 7.24 51.4 48.6
2nd. HE 189 3.93 51.3 48.7
Total 4,741 98.63 52.8 47.2
Missing 66 1.37 — —
Total 4,807 100.00 53.0 47.0
Revista de Psicodidáctica, 2014, 19(1), 67-91
ATTITUDES TOWARDS MATHEMATICS:
CONSTRUCTION AND VALIDATION OF A MEASUREMENT INSTRUMENT 73
Variables and instruments prior works in this type of meas-
urement instruments such as those
To construct the Escala de of Pietsch, Walker, and Chapman
Actitudes hacia las Matemáti- (2003).
cas (EAM; in English, the Scale All these questions were as-
of Attitudes towards Mathemat- sessed by experts in Didactics of
ics), we drew from the works re- Mathematics. Through these as-
viewed in the previous section sessments, the most pertinent ques-
that present five generalized fac- tions due to their relevance (the
tors: liking-enjoyment of math- items must be clearly related to the
ematics, anxiety towards math-
object of study) and clarity (sim-
ematics, perception of difficulty,
ple, easily understood statements)
perceived utility, and mathemati-
were selected. A pilot study on a
cal self-concept were the thematic
small sample with this selection
fields chosen to prepare the initial
was carried out. After the elimina-
items of the test.
tion and/or selection of the most
First, a broad set of questions
adequate items, we prepared the
related to these five factors was
final scale made up of a total of
designed. To assess the factors
37 questions, which are presented
associated with liking or enjoy-
grouped by factors in Table 3. In
ment of mathematics the Enjoy-
this final scale, all the items are
ment of Mathematics subscale of
rated according to the degree of
Aiken (1974) and the Liking scale
agreement with the statement on a
of Fennema and Sherman (1976)
5-point Likert-type scale (values
was used. To select the questions
ranging from 0 to 4).
related to anxiety towards math-
ematics, we drew from the works
Procedure
of Richardson and Suinn (1972).
To measure the perception of dif-
The scales were administered
ficulty of mathematics, we drew
by the authors and collaborat-
from the previously cited works
ing teachers during the academic
of Aiken (1974) and Fennema and
Sherman (1976). The questions courses 2009/2010, 2010/2011
of the factor of perception of util- and 2011/2012. The scales were
ity of mathematics were devel- anonymous and completed by the
oped based on the proposals of subjects of the sample in the pres-
Aiken (1974) and Fennema and ence of the teacher and/or collab-
Sherman (1976). The design of orator. Before collecting the data,
the questions related to the per- both the parents’ informed con-
ception of efficacy and/or compe- sent and the authorization of the
tence in mathematics (mathemat- headmasters of the schools were
ical self-concept) was based on obtained.
Revista de Psicodidáctica, 2014, 19(1), 67-91
74 ANDRÉS PALACIOS, VÍCTOR ARIAS, AND BENITO ARIAS
Results ods obtained comparable results.
Both analyses were carried out
Exploratory factor analysis on the polychoric correlation ma-
trixes, in view of the ordinal na-
The original sample ture of the input data. The ad-
(n = 4,741) was divided into two equacy of the input data was
randomly extracted subsamples confirmed by means of Bartlett’s
(n1 = 2,371 and n2 = 2,370). The sphericity test, the KMO index,
first one was used to carry out the and the matrix determinant (Ta-
exploratory factor analysis (EFA) ble 3).
and the second was used as a vali- Both the pattern matrix and
dation sample for the confirma- the structure matrix obtained sim-
tory factor analysis (CFA) and for ilar results in the first subsample;
the analyses based on the item re- the items presented a practically
sponse theory, described below. identical loading distribution in
Three items were eliminated (“I the different factors. This similar-
am happier about getting a 10 in ity was corroborated by means of
mathematics than in any other sub- the Pearson correlations and the
ject”, “My parents are more con- congruence coefficients. Table 2
cerned about my outcomes and shows that the Pearson correla-
grades” and “When I have trouble tions reached a mean of .991, rang-
with mathematics, I usually ask ing from .982 to .996. The congru-
for help from my family”) because ence coefficients calculated from
they presented corrected homoge- the pattern matrix loadings ranged
neity indexes lower than .20. The from .987 to .997 (exceeding the
distributions of the variables age limit of .95, habitually considered
and sex were similar in both sub- acceptable for this type of analy-
samples. The standardized Pear- sis). Similar results were obtained
son residuals ranged from –1.09 to when comparing the matrixes of
1.03, and the model [AGE, SAM- the two random subsamples using
PLE][SEX] was nonsignificant the PAF extraction method. The
(χ2 = 11.859, p = .690), so the Pearson correlation coefficients
(15)
hypothesis of equivalent subsam- ranged from .901 to .986, and the
ples was accepted. congruence coefficients from .955
Exploratory factor analysis to .997.
was carried out with the SAS, v. PROMAX oblique rotation
9.2 program. When determining was used, because prior research
the factor structure of the EAM, indicated that the dimensions of
we used two extraction proce- anxiety towards mathematics are
dures (Principal Axis Factoring, correlated (e.g., Pajares & Miller,
PAF, and Maximum Likelihood, 1994).
ML) to verify whether both meth-
Revista de Psicodidáctica, 2014, 19(1), 67-91
ATTITUDES TOWARDS MATHEMATICS:
CONSTRUCTION AND VALIDATION OF A MEASUREMENT INSTRUMENT 75
Table 2
Pearson Correlations and Congruence Coefficients
Extraction methods Random subsamples
r CC r CC
F1 .994 .997 .986 .997
F2 .996 .997 .980 .994
F3 .992 .994 .962 .985
F4 .982 .987 .901 .955
The results of the EFA pre- I do, I always get low grades in
sented in the following para- mathematics”). As the five- factor
graphs correspond to the PAF ex- structure is not statistically justifi-
traction method, in view of the able, an optimized parallel analy-
more ‘classical’ nature of this sis (Timmerman & Lorenzo-Seva,
method compared to the ML 2011) was conducted by comparing
method (Pett, Lackey, & Sullivan, the Eigenvalues obtained through
2003, p. 103). analysis with the randomly gen-
To determine the number of erated Eigenvalues of 1000 sub-
factors to retain, various criteria samples obtained from the original
into account were taken: the Kai- sample. This analysis is currently
ser-Gutman rule, Cattell’s scree- considered the most adequate to
test and parallel analysis. The Kai- make decisions about the number
ser-Guttman rule (Eigenvalues of factors to retain (Hayton, Al-
higher than 1.00) suggested retain- len, & Scarpello, 2004). As of the
ing five factors, as did the results fourth factor, the magnitude of the
of the scree-test. However, we de- randomly generated Eigenvalues
cided to disregard this recommen- exceeded that of the Eigenvalues
dation because both methods usu- obtained through the analysis, so
ally lead to overfactorization; in we decided to retain the four-factor
fact, the fifth factor presented an solution.
Eigenvalue of only 1.10, it explains In accordance with the conven-
less than 3% of the common vari- tional criteria in this type of analy-
ance, and only includes two items sis, there were three item-retention
in a dimension that could be called criteria: (a) the item loading on the
Learned helplessness (“Except for main factor should at least reach
a few cases, no matter how much the value of .40; (b) the loading on
effort I put out, I cannot understand the remaining factors should not
mathematics” and “No matter what exceed the value of .35; and (c) the
Revista de Psicodidáctica, 2014, 19(1), 67-91