In the last phase of the present study we used hierarchical linear models (HLM) to explore further the connections between the affective factors and students’
performance in mathematical literacy. These models show how strongly each of the four affective factors explored in this study were connected to students’
performance, when they were studied in combination and not individually. In addition to the affective factors, gender was also used as a variable in the models. A benefit of using HLM was that it allowed us to take into account the structure of the school data: Students are clustered in schools and in the case of the OECD model schools are further clustered in countries. If this structure is not taken into account, the results could be incorrect because, for example, the results of students from the same school tend to be more similar than the results of students from different schools (the so-called intra-school correlation) (Malin, 2005).
Because of the structure of the data all factors were studied at two levels in the Nordic countries: School level contextual factors were computed by averaging the student level values of the original factors. In the OECD model the country-level was used as an additional third level and factors for this level were computed by averaging the school level factor values.
The model descriptions in Table 5 show all factors that remained statistically significant in the country or OECD average analyses. The model descriptions include only student and school level factors. This means that for the OECD model none of the country-level factors were statistically significant predictors of student achievement. When the factor coefficients are studied, it should be borne in mind that if other variables are added to the model, the coefficients for the present factors would most likely change.
In any case, according to the descriptions the affective factors were quite similarly connected to performance in the Nordic countries. At the student level, self-concept in mathematicshad the strongest connection with performance. An increase of one
point on the self-concept scale was connected to an increase of about 40 points in mathematics performance in the Nordic countries. The effects of gender, interest and enjoyment, and anxiety were clearly smaller, and in Denmark and Iceland instrumental motivation was not statistically significantly related to students’
performance. Surprisingly, interest in and enjoyment of mathematics were negatively connected with performance and the coefficient for boys was also negative.
At the school level anxiety in mathematicsin particular was statistically significantly connected with performance in the Nordic countries. Instrumental motivation and self-concept were also statistically significant predictors of performance at the school level in Denmark and Finland, respectively.
These simple models explained about one-third of the total variance (variation in the students’ results) in the Nordic countries, the proportion explained varying from 29.9% in Sweden to 38.3% in Finland. Proportions of this size are quite typical for this kind of model (Kupari & Törnroos, 2004). However, the
Denmark Finland Iceland Norway Sweden OECD Intercept (factor values 0) 517.7 550.9 528.4 504.6 514.7 516.5 School level factors
Gender (proportion of boys) -42.4
Interest and enjoyment -27.6
Instrumental motivation -30.2
Self-concept 26.6 20.3
Anxiety -33.2 27.5 -14.7 -30.3 -49.4
Student level factors
Gender (student is a boy) -6.9 -15.7 -25.3 -16.8 -11.5 4.9
Interest and enjoyment -13.8 -10.3 -8.0 -4.1 -11.0 -7.7
Instrumental motivation 5.0 8.3 6.0 3.1
Self-concept 40.3 44.7 40.9 36.2 40.7 27.0
Anxiety -19.7 -10.3 -7.4 -14.7 -16.7 -12.4
Variation explained (%):
Country level (OECD model) 27.8
School level 28.7 45.6 27.8 52.8 31.4 26.7
Student level 33.7 37.9 31.5 36.8 29.7 34.3
Total 33.0 38.3 31.4 37.9 29.9 31.5
Reported effects are statistically significant with p<0.05.
Table 5 Hierarchical linear models for the Nordic countries and OECD countries on average
proportions of school-level variance explained were surprisingly high in Finland and Norway.
How should the coefficients of the model be interpreted? When these kinds of model descriptions are presented the interpretation of the results is at least as important as the description of the model. For these models in particular two results need to be discussed: Gender differences and the role of interest in and enjoyment of mathematics. The gender coefficients (boys compared to girls) of the Nordic models are all negative. This does not mean that boys performed worse than girls in PISA mathematics tasks. In fact, boys had better results than girls in all Nordic countries except in Iceland. What these coefficients do mean is that if a girl and a boy have similar values for the affective factors, the boy has usually performed worse than the girl. Because girls reported, for example, lower self-concepts and interest in mathematics than boys in the Nordic countries, one way to improve girls’
performance could be to strengthen these aspects of girls’ attitudes to mathematics.
The coefficients for interest in and enjoyment of mathematicswere indeed
unexpected. Interestitself was positively associated with students’ performance in all Nordic countries (figure 1), but when it was combined with self-conceptin the models, the coefficients became negative. This may imply that the self-concept index in PISA also accounts, to a great extent, for the positive association of interest and enjoyment with performance. The negative coefficients for interest and
enjoymentin the model may be caused by, for example, the fact that many students with a high self-concept and good results are not interested in mathematics. And, at the other end of the scale some students are very interested in mathematics but have great problems with learning it.
At the school level, anxiety was the factor that was most often associated with students’ performance. This means that the anxiety felt towards mathematics varies between different schools and this variation is reflected in students’ mathematics results. In Denmark, Norway and Sweden the effect was negative, as expected. That is, the more students felt anxiety at school the lower were their results. In Finland, however, the effect was reversed. Students’ results were higher in schools where they felt more anxiety. This unforeseen result may be due to a combined effect of anxiety and self-concept at the school level, similar to the combined effect of interest and self-concept at the student level previously discussed. A similar unforeseen effect is seen in the Danish results, where instrumental motivation was negatively associated with performance at the school level.
Because the OECD average model had no statistically significant country-level factors, it looked quite similar to the Nordic models. The fact that no country-level
factors are sustained in the final models is in itself an interesting result. It means that in this model the differences between the OECD countries were less important than the differences between schools and students within the individual countries.
Some differences between the OECD model and the Nordic models are worth discussing. In the OECD average model, students’ self-concept in mathematics was not as strong a predictor of performance at the student level as in the Nordic countries.
It is also interesting that in the OECD model, boys had better results than girls even though their affective values were the same. In the OECD model gender also had an interesting effect at the school level: According to the model, students in classes with more girls got better results than students in classes with more boys.
In the OECD model, interest and enjoyment had a negative effect on performance both at student and school levels. This seems to validate the findings from the Nordic models but the reasons behind these findings need further research. In the OECD model, anxiety had a noticeably stronger negative effect on performance at the school level than in the Nordic countries. When all OECD countries were considered, the variation in anxiety at the school level was much greater than in the Nordic countries, which were all among the countries where students reported the least anxiety towards mathematics (OECD, 2004). In the OECD model, this increase in variation was also reflected in performance at the school level.