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## Estudios pedagógicos (Valdivia)

*versión On-line* ISSN 0718-0705

### Estud. pedagóg. v.31 n.1 Valdivia 2005

#### http://dx.doi.org/10.4067/S0718-07052005000100006

Estudios Pedagógicos XXXI, N 1: 97-112, 2005
TIMSS & PIRLS International Study Center, Boston College, 188 Beacon Street, Chestnut Hill, MA 02467, USA. ramiremb@bc.edu ; ramirezmjose@yahoo.com
Using data from the 1999 Trends in International Mathematics and Science Study (TIMSS), this study reveals that the Chilean students like and value mathematics, but have an inflated self-perception of their mathematic competence. Hierarchical linear models were used to predict mathematic achievement both at the student-within-class and class levels. At both levels, students' perceived difficulty of doing mathematics, expectations for further education, and beliefs regarding the causes of their mathematics outcomes were significant predictors of achievement. However, classes having more students liking mathematics had significantly lower mean scores. This later result is interpreted as the consequence of the more demanding curriculum and tougher grading standards used in higher performing classes.
Usando datos del Estudio Internacional de Tendencias en Matemáticas y Ciencias de 1999, este trabajo muestra que los alumnos chilenos disfrutan las matemáticas y las encuentran importantes. Al mismo tiempo, los jóvenes tienen una inflada autopercepción de sus habilidades matemáticas. En este estudio se usaron modelos lineares jerárquicos para predecir rendimiento en matemáticas a nivel de alumnos y a nivel de las salas de clases. En ambos niveles, fue significativo el efecto de dificultad de hacer matemáticas, creencias en torno a las causas del desempeño en esta disciplina y expectativas de educación futura. Paradójicamente, en
In Chile, the debate about how to improve academic achievement usually points to structural factors such as teacher education programs, school resources, or curricula. However, little attention has been given to students' characteristics such as how much they like mathematics, how important they think it is, how difficult they perceive it is to do well in mathematics, and what factors they believe affect their mathematics performance. By the same token, little is known about students' aspirations for further education: Do Chilean students look forward to finishing their secondary school or do they plan to enter the university? Students' opinions and beliefs regarding mathematics, how much they like and value it, and what they forecast for their own future education can all be understood as different facets of The study of attitudes in the school setting has several complications, though. Because of the different facets of the attitude construct, what is meant by attitudes toward mathematics varies from one study to the other. Moreover, it is common to find studies that do not use the term attitudes, but whose focus lay in one or more of its facets (e.g., academic self-perception, locus of control). The unit of analysis is another complication in the study of attitudes. Relationships that may hold true at the student level may not be of the same nature and strength at the school or classroom level. From the Trends in International Mathematics and Science Study (TIMSS) 1995 there is evidence that the majority of 8 Using data from the Chilean national assessment system, SIMCE, Ramírez (2003) found that 4 In Israel, Nasser and Birenbaum (2004) studied the relationship between mathematics achievement and some learner-related variables, including self-efficacy, beliefs regarding knowledge, and attitudes toward mathematics. The authors conducted their study in two samples of 8 These authors operationalized attitudes towards mathematics as encompassing how much the students liked mathematics, how difficult they found it, and how important they thought it was. The relationship between mathematics achievement and attitudes was not so clear. They found that attitudes had minor and insignificant effects on mathematics achievement of Jewish children, while it had a modest but significant effect on the Arabs. Arab students reporting more positive attitudes toward mathematics attained a higher performance than those reporting more negative attitudes. Interestingly, while the Jewish students outperformed their Arab peers in mathematics, the latter reported more positive attitudes toward this subject. Shen and Pedulla (2000) reported that more students from lower performing countries (or systems) have more positive attitudes toward mathematics: they found it easier and perceived themselves as more capable of doing well in this subject. These authors interpreted their findings as the consequence of more demanding curricula and high standards used in high achievement countries, which may lead the students to see mathematics as a harder topic, and themselves as not so strong in doing mathematics. Benham (1995) reviewed studies relating students' self-perceptions and academic achievement. When students believed that their academic performance was the consequence of their own actions (e.g., studying hard, perseverance, motivation) rather than the consequence of factors out of their control (e.g., good luck, innate ability), they had better academic performance. In the literature, the former are said to have an internal locus of control, while the latter have an external one. Stemler (2001) provided further evidence of the importance of the students developing an internal locus of control. From an analysis of 4 In education, it is a fact that the socio-economic background of the students' families plays an important role in explaining academic achievement. Poorer students attain substantially lower achievement levels than their peers from more economically advantaged backgrounds. In Chile, the educational system is highly stratified by social class. Broadly speaking, public schools serve the poorer families, private-subsidized schools serve the middle-class, and the elite-paid schools serve the richer families. Not surprisingly, the public system has consistently gotten lower scores in standardized achievement tests (Ministerio de Educación [Mineduc], 2001). In this scenario, attempts to account for the mean achievement differences among classes from different schools should consider the effect of the socio-economic level of the communities served by the schools. Within the Chilean schools, though, students within communities are homogeneous in their socio-economic background. Hence, this factor cannot account for the large variations in achievement observed within the classes. In this context, students' attitudes toward mathematics may account for an important proportion of the achievement variation among classmates. The purpose The following research questions guided the analyses:
My interest in doing this study comes from the firm belief that structural changes are a necessary but not sufficient condition to reach the goals set by an ambitious educational reform in Chile. After a decade of improvements in areas like libraries, curriculum, and working conditions for teachers, there is no evidence of improved students' achievement. Perhaps it is time to put more emphasis on the students themselves and on how their characteristics may be affecting their own learning processes. The study of students' attitudes toward mathematics and their link to academic performance is definitely an aspect worth examining.
For the multi-items factors, indices that combined information from several questions were created with the purpose to increase the validity and reliability of the measures, and also to allow for data reduction. Items conceptually related to the constructs, with factor loadings The indices were computed averaging the points associated with each response option. Before averaging, all the questions were scaled in the same direction. In case of missing data, the index value was computed based on the valid responses to the other source questions. This allowed maximizing the number of valid cases to be used in the analyses. For the single-item factor A socio-economic index (SEI) was used as an indicator of the average cultural and economic capital of the students' families. This index was computed averaging the score points associated with three questions: "How far in school did your mother and father go?," "about how many books are there in your home?" and "do you have any of these items at your home?" (e.g., computer, refrigerator). The mean, standard deviation, and internal consistency (Cronbach's a) for mathematics achievement (outcome), liking mathematics, importance of mathematics, difficulty of doing mathematics, importance of luck and talent, students' expectations for further education, and the socio-economic index are presented in Table 2. While reliability is not high, it is important to bear in mind the reduced number of items used as indicators of each index.
Despite the fact that all the factors were significantly correlated with the outcome ( The inter-correlations between the predictors were of low to moderate magnitude; however, they were significant in all but one case at
In the TIMSS sampling design, all the students from one intact class participated in the study, and only one class per school was sampled. Because of this nested design, the final sampling units (students) were not independent one of each other, and so hierarchical linear models (HLM) were statistically appropriate to model mathematics scores. This technique also presented the advantage of providing a more comprehensive picture of the strength of the relationship between factors and achievement at different levels of analysis. At the student-within-class level, the sub-model predicted differences in mathematics scores among classmates who shared the same classes and schools, and students' attitudes variables were used as predictors. At the class level, the sub-models predicted differences in mean classes scores, and the aggregated students' characteristics were used as predictors. The mathematics scores (five plausible values) were used as the outcome variable, while the socio-economic index served as a covariate at the class level. The following equations specify the sub-models used in the hierarchical linear models. The unconditional model informed about the proportion of variance that lay among classes and within classes. In this model no predictors were entered at either level (Equations 1 and 2):
In the
In the
Table 3 shows the questions related to attitudes toward mathematics used in this study, together with the percentage of students by response category and their mean mathematics score. In all the cases, a one-way analysis of variance (ANOVA) showed that mathematics achievement varied significantly depending upon the response categories of the questions ( Some noteworthy results were that 51% of the Chilean students endorsed (agreed or strongly agreed) the statement "to do well in mathematics you need good luck," and 73% did so with "to do well in mathematics you need lots of innate talent." Despite the poor performance of the Chilean students in the TIMSS mathematics test The vast majority of the students (81%) strongly agreed Figure 1 represents the relationship between mathematics scores and the degree of agreement with "to do well in mathematics you need good luck." There is a monotonic relationship between the variables: increases in one are associated with increases in the other. Students who strongly disagreed with the statement "to do well in mathematics you need good luck" scored 85 points more (one standard deviation) than their peers who strongly agreed with it. Figure 2 shows how mathematics scores increased for students with higher expectations for further education (not shown in previous table). Students aspiring to finish the university (43%) averaged 429 points on the mathematics test, compared to the 355 points obtained by their classmates aspiring to finish secondary school only (15%). HOW MUCH ACHIEVEMENT VARIANCE CAN BE PREDICTED BY STUDENTS' ATTITUDES TOWARD MATHEMATICS? The HLM unconditional model indicated that 61% of the variations in mathematics score lay within the classes while 39% lay among the classes from different schools. Table 4 shows the results of the hierarchical linear models at the student-within-class level; in this model, the five attitudes factors were entered as predictors. This model accounted for 23% of the outcome variance that lay within the classes. Significant predictors were difficulty of doing mathematics, importance of luck and talent, and expectations for further education. Within the classes, students reporting that doing mathematics was more difficult had lower scores than their mates reporting that this subject was not so difficult ( At the class level (Table 5), the socio-economic index was entered alone in Step 1; this control variable was highly significant, explaining up to 68% of the mean score differences among the classes. Classes with more students from economically advantaged homes had higher mean scores than classes with less advantaged students. In Step 2, another sub-model encompassing both the socio-economic index and the aggregated attitudes variables was tested. This sub-model accounted for 80% of the outcome variance (i.e., 12 percentage points more than the previous model). With the exemption of importance of mathematics, all the predictors made a significant partial contribution to the outcome variance ( Liking mathematics made a negative partial contribution to the outcome (
Consistent with the international evidence (Kifer, 2002), the Chilean 8 Nevertheless, less auspicious results arise from the analysis of other aspects of students' attitudes toward mathematics. Half of the students think that to do well in mathematics they need good luck, while 40% think that they need an innate talent that they do not have. It is clear that these students do not feel in control of their own learning processes. Consequently, it is hard for them to see a connection between their own efforts and motivation to do well and their academic performance. These findings suggest failures in molding a more active role for the students in their own learning processes. In Chile, there is a recognized gap between the objectives stated in the national curriculum and what the students know and are able to do (Mineduc, 2001). This country was also among the lowest performing in TIMSS 1999 (35 Mathematics difficulty and self-competence are mainly a function of the difficulty level of the topics covered in classes and, most importantly, of the grades the students get in classroom evaluations. There is evidence that the Chilean teachers cover little advanced mathematics topics in the 8 It is highly likely that the students use their own grades in judging how well they do in mathematics. In Chile, it is well known that the grades do not reflect the curricular standards; grade inflation is a pervasive problem (Riveros, 2004). Teachers "lower the bar" to the average student, and grade them based on this relative standard. Too many schools work in a culture of low academic standards and poor expectations for their students. This lack of rigor is perceived as one of the main problems of the Chilean education system (Arancibia, 1994). This situation should lead to a serious review of the teachers' assessment and evaluation practices. If the students are not corrected in their errors, and if they do not know their real level of performance, it will be difficult to put more pressure on them to study harder. By the same token, it will be difficult to involve parents in supporting their children's learning. The research questions of this study asked how well the students' attitudes toward mathematics predict the achievement differences among classmates, and how well they predict the achievement differences among the classes. Within the classes, the students' attitudes accounted for almost one-fourth (23%) of the variations in achievement, which is a substantial amount. Students who rely less on factors out of their reach (e.g., good luck, innate talent), report little difficulty with mathematics, and expect to finish the university, attain significantly higher achievement than their classmates who rely more heavily on luck and talent, think that doing mathematics is harder, and do not aspire to finish the university. At the class level, students' attitudes toward mathematics accounted for one-tenth of the variations in mean class achievement, beyond the effect of socio-economic level. Classes having more students expecting to finish the university, less students perceiving that doing well in mathematics is difficult, and less students who rely on good luck and innate talent to do well, attain higher mean achievement than classes with less students planning to go into the university, more students perceiving that doing mathematics is hard, and more students relying on good luck and innate talent. These findings can be interpreted as a positive cycle where expectations for further education, perceived self-efficacy of doing mathematics, and mathematics achievement have mutual influences one on each other. The findings also suggest the importance of fostering a perception that students' academic outcomes depend to a greater extend on students' own effort and motivation to do well, and less on factors out of their control, such as good luck or innate ability. Timely feedback, assessments that reflect the real students' performance levels, and opportunities to compare previous to current performance levels are strategies that may help the students to change their beliefs regarding the causes of their mathematics outcomes (for a detailed discussion, see Benham, 1995). Contrary to the popular belief, liking or valuing mathematics has little further to do with achievement, when analyzed at the student level. This assertion makes sense considering that almost all the Chilean students like and value mathematics. A measurement problem may be obscuring the link of these factors with achievement. It is perhaps the case that students have a tendency to answer according to what is socially accepted and valued (social desirability), hence that the vast majority of the students report liking and valuing mathematics. Maholmes (2001) speculates on this and other plausible explanations of why minority students in the United States report liking mathematics. Interestingly, while liking mathematics was not a significant predictor of achievement within the classes, it has a significant and negative relationship with the outcome at the class level: classes with higher mean achievement have fewer students who like mathematics. As already suggested by other authors (Shen & Pedulla, 2000), it may be the case that students in higher performing classes do not like mathematics as a consequence of the more demanding curriculum and stringent grading policies. On the contrary, less demanding classes may be promoting positive affects at the cost of higher achievement; thus, reaching learning levels far below the curricular expectations for the grade. It is reasonable to conclude that, in Chile, it is important to change some students' beliefs regarding the nature of learning and the factors that affect it. The development of a more realistic appreciation of the students' own capacities to do well in mathematics also seems desirable. It is likely that changing students' self-appreciation will require improving teachers' grading standards. These findings are encouraging since they point toward variables that are relatively independent of the strong effect of socio-economic level. However, they are also challenging since changing beliefs, opinions, and evaluation standards (toward self and others) imply changing some cultural patterns deeply rooted in the Chilean society.
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