The next step consisted of applying cluster analysis to see how countries group together according to similarities in their data for the meta-constructs numbered 1-11 above. Countries are clustered according to similarities in what is going on in math classrooms as well as students’ attitudinal and cognitive responses thereto.
Cluster analysis is a method which allows us to systematically investigate similarities and differences between countries in the data for the actual meta-constructs. The idea is to quantify the similarities between each pair of countries by, for example, applying ordinary (Pearson) correlation coefficients. There are many alternative measures of similarities, e.g. squares of differences between the two countries summed over all meta-constructs at hand. Strong similarities between countries mean that there is a tendency for them to report high and low values for the same meta-constructs, i.e. the countries have similar patterns from variable to variable. On the other hand, consistently low or high values for a country do not influence correlations.
The analysis starts by combining the two countries with the highest similarity into one group. At the next step two other countries are clustered together, or another country is combined with a group that has already formed, depending on which similarity is the largest. Once a group is formed, its group mean for each meta-construct will then be used to calculate its similarities with individual countries or other groups.The process goes on until all countries have finally been combined into one large group.
Figure 1 is a dendrogram, which depicts how countries are clustered into increasingly larger groups going from left (high similarity) to right (low similarity).
Countries with similar data tend to appear as neighbours in the country list and combine vertically relatively ‘’early’’ (to the left) in the dendrogram. A long line of arrows before another country merges with a group reflects what is often referred to as high external isolation for the group. As mentioned above, there are many possible measures of similarity other than correlations that could be applied to the data (see for instance Olsen 2005a). By applying a series of relevant but slightly different criteria, some features appear across the versions, and we will focus on some of these in further analyses. On the other hand, there are also differences between the various versions, so one should be careful not to pay too much attention to the detail. We want to emphasise that our discussion in the following section does not relate solely to figure 1.
Figure 1 Dendrogram of countries based on correlations of meta-constructs
Based on figure 1 (and also on a number of alternative versions) we have suggested meaningful country clusters for further analysis. In order to do this we applied the following guiding criteria: At least four and less than eight countries should be in each group. Further, the calculated reliability (as measured by Cronbach’s alpha) should be above 0.75, and each member of the group should contribute positively to alpha. And most importantly, each grouping should be conceptually meaningful in the sense that the group can be labeled in a way that provides some explanation of which countries are included and which ones are not.
Table 1 displays the proposed country groups together with measures of reliability and average (as well as minimum and maximum) correlation coefficients between countries within the group. As can be seen from the table, the groups do generally meet the criteria above. In our view there are remarkable patterns in the meta-construct data that lead to these meaningful country groups. However, there are some important comments to make at the start. Firstly, we are primarily concerned with outcomes for the actual Nordic countries. The group that for convenience is labeled ‘’Nordic’’ differs in important respects from the actual Nordic group. More will be said about this group later. Secondly, the ‘’English-speaking’’ group does not readily fit our criteria, since the four selected countries do not show a strong tendency to cluster early in figure 1, as they do in reading, mathematical and science achievement. It is also worthwhile noting that the European English-speaking countries Great Britain and Ireland do not fit into this group. Thirdly, for the ‘’East Asian’’ group there is a strong tendency towards two strong pairs of countries: Japan–Korea and Hong Kong–Macao, respectively. This is no surprise, given the fact that the latter two are not countries, but semi-independent provinces within the same country, China. Finally, in spite of their close linkages in figure 1, group 7 countries (France, Italy, Belgium and
Luxembourg) are not clustered very strongly, as is seen both from the low alpha and the low minimum correlation (between Belgium and France). Even labeling this group is difficult; ‘’French’’ is used for convenience.
It should also be emphasised that 11 of the 41 countries, among them Iceland and Denmark, tend to remain isolated or to form pairs between two clusters, or they tend to combine with countries which conceptually do not seem to belong to the same group. Therefore, according to the criteria, these 11 countries have not been included in table 1.
Table 2 displays the characteristics for each of the seven groups of countries in table 1. The standardised values are shown for all meta-constructs. The extreme absolute values shown in bold indicate the most pronounced characteristic features.
From table 2 it can clearly be seen that each group has a distinct pattern that is different from that of the others. Group 1 (Less developed countries) is a
particularly interesting case, since most of the values are extreme. Achievement score and home backgroundare both very low. More surprising probably are the very high values for Teacher support, Subject motivation, Learning strategiesand Social
motivation. Group 5 (Eastern Asia) also stands out as having many extreme values, in particular for the same factors as group 1, but in most cases with the opposite sign,. The four European Groups (2, 4, 6 and 7) have relatively more in common, and the same is true of groups 3 and 5. Accordingly, the data provide evidence for the fact that the seven country groups can conceptually be ordered into the following three ‘’meta-groups’’ of countries:
A. Group 1, Less developed countries
B. Groups 3 and 5 combined, English speaking+ East Asia C. Groups 2, 4 and 6 combined, Europe
Country group Countries Cronbach’s Average and range alpha of correlations 1. Less developed Brazil, Mexico,
Indonesia, Thailand,
Tunisia, Turkey 0.94 0.74 (0.52 – 0.97)
2. “Nordic” Finland, Netherlands,
Norway, Sweden 0.93 0.76 (0.65 – 0.84)
3. English speaking Australia, Canada,
New Zealand, USA 0.76 0.50 (027 – 0.80)
4. East Europe Poland, Slovak rep.,
Czech rep., Hungary 0.86 0.65 (0.46 – 0.82) 5. East Asia Hong Kong, Macao,
Japan, Korea 0.84 0.61 (0.32 – 0.91)
6. German speaking Switzerland, Liechtenstein,
Austria, Germany 0.95 0.82 (0.73 – 0.93)
7. “French” France, Belgium,
Luxembourg, Italy 0.72 0.46 ( 0.14 – 0.79) Table 1 Groups of countries with labels, reliability and correlation measures. 30 of the 41 countries appear in the groups
The order of the factors indicates that A and C lie at each end of the spectrum, with B somewhere in between. This over-arching structure can also be seen directly from figure 1.
It is very interesting to compare our findings so far with the patterns that emerge when countries are clustered based on scores for individual achievement items. In the chapter by Olsen (in this volume), groups of countries are established which are essentially the same as those presented here. It appears remarkable to us that that the patterns are so similar, given the fact that they are based on totally different types of data. Together the two sets of analyses mutually reinforce each other in the sense that there seem to be common educational features among groups of countries which may be based on deep underlying traits linked to historical and philosophical traditions, geographic patterns and linguistic influences.