Vani Mark Francis Bailey
email: mf.bailey@ulster.ac.uk

Vani K Borooah
email: vk.borooah@ulster.ac.uk

Ability? A Cross-Country Analysis Based on Test Scores of 15-year Olds

Keywords: Mathematical Ability, Inequality, Countries, Home, School

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Introduction

Economists are concerned with the levels of mathematical ability that individuals have because the mathematical ability of individuals affects economic performance. At a macroeconomic level, studies making use of international school test score data have found a link between mathematical ability and the economic growth of the state (Hanushek and Kimko, 2000). This linkage has been recognised by policy makers.

For example, Alan Greenspan giving testimony before the Committee on Education and the Workforce of the U.S. House of Representatives in March 2004 noted that: "Research on wealth creation in both emerging and developed nations strongly suggests that it is the knowledge and the skill of our population interacting under our rule of law that determine our real incomes" And this was partly due to the fact that: "Many of our students languish at too low a level of skill, and the result is an apparent excess of supply relative to a declining demand. … The hypothesis that we should be able to improve upon the knowledge that our students acquire as they move from kindergarten to twelfth grade gains some support from international comparisons. A study conducted in 1995 revealed that, although our fourth-grade students were above average in both math and science, by the time they reached their last year of high school they had fallen well below the international average." (Greenspan (2004)) At a microeconomic level, studies have found positive wage returns for individuals to mathematical ability (Kenny et. al., 1979, Dolton and Vignoles, 2000, McIntosh and Vignoles, 2000). Dolton & Vignoles (2000) refer to the returns to an individual due to the possession of Mathematics 'A'-level (a Mathematics qualification taken at post-16 level); interestingly they find that that the return is not

Equity Sensitive Indicators of Student Assessment Scores

In a paper prepared for the 1995 Human Development Report, Anand and Sen (1997) pointed out that a country's non-economic achievements were likely to be unequally distributed between subgroups of its population: for example, in terms of gender equality, which was the focus of their concern, the female literacy rate, or female life expectancy, was often lower than that for males. In the face of such intergroup inequality, they argued that a country's achievement with respect to a particular outcome should not be judged exclusively by its mean level of achievement (for example, by the average literacy rate for a country) but rather by the mean level adjusted to take account of inter-group differences in achievements. Anand and Sen (1997) proposed a method, based on Atkinson's (1970) seminal work on the relation between social welfare and inequality, for making such adjustments and they termed the resulting indicators equity sensitive indicators. They further suggested that assessments of country achievements should be made on the basis of such equity sensitive indicators rather than, as was often the case, on the basis of its mean level of achievement. This would, then, allow a comparison between two countries, one of which had a lower mean achievement level, but a more equitable distribution of achievement, than the other. 1 In this section we apply these ideas to the student assessment scores (SAS) -hereafter, simply, "scores" -in the PISA data. Suppose that X is the average score in a country where i X is the score of student i (i=1…N). We know that, because of inequality in the distribution of scores between students, the average score of a country will not be achieved by all its students.

1 Anand and Sen (1997) compared the Honduras (with an average literacy rate of 75%, distributed between men and women as 78%, 73%) with China (with an average literacy rate of 80%, distributed between men and women as 92%, 68%) and asked which country should be regarded as having the "better" achievement with regard to literacy: China with a higher overall rate or the Honduras with greater gender equality? Therefore, in assessing the SAS achievement of a country, by how much should we reduce its average SAS to take account of inequality in scores?

The answer to this question depends on how averse we are to inequality. In his seminal paper on income inequality, Atkinson (1970) argued that we (society) would be prepared to accept a reduction in average income, provided the lower income was equally distributed, from a higher average income which was unequally distributed. 2The size of this reduction depended upon our degree of "inequality aversion" which Atkinson (1970) measured by the value of a (inequality aversion) parameter, 0 .

When 0 = , we are not at all averse to inequality implying that we would not be prepared to accept even the smallest reduction in average income in order to secure an equitable distribution. The degree of inequality aversion increases with the value of : the higher the value of , the more averse we would be to inequality and, in order to secure an equitable distribution of income, the greater the reduction in average income we would find acceptable.

These ideas can, equally well, be applied to student assessment scores. We can reduce the average score, X , of a country by the amount of inter-student inequality in scores to arrive at e X , a "group equity sensitive" score for the country, e X X . We refer to e X as the equally distributed equivalent score. The size of this reduction (as given by the difference, e X X ) depends upon our aversion to inequality: the lower our aversion to inequality, the smaller will be the difference and, in the extreme case in which there is no aversion to inequality ( 0 = ), there will be no difference between the average and the equity sensitive scores. Three special cases, contingent upon the value assumed by , may be distinguished: ( )

1/N 1 < N N e i i X X X = = .
3. When 2 = , e X is the harmonic means of the student scores:

1 1 N e i i N X X X = = < .
Table 1 shows the equity sensitive scores for each of the 41 countries in the PISA data, contingent upon the amount of inequality in the distribution of scores between all the 15 year olds in (the sample for) that country.

Inequality Decomposition

Suppose that the N students are sub-divided into K mutually exclusive groups (indexed k=1..K) where group k has N k students and a score vector, S k . An interesting question is how much of the observed difference between students in their assessment scores is the result of differences between groups and how much is the result of differences within groups? We answer this question using additively decomposable inequality indices.

An inequality index ( ; ) I N S is said to be additively decomposable if:

1 ( ; ) ( ; ) K k k k k I N I N w = = + = + S S B A B
where: ( ; ) I N S represents the overall level of inequality defined over the vector of all the scores, S; ( ; )

k k I N S
represents the level of inequality within group k; Aexpressed as the weighted sum of the inequality in each group, w k being the weightsand B represent, respectively, the within-group and the between-group contribution to overall inequality. If, indeed, inequality can be 'additively decomposed' along the above lines then, as Cowell and Jenkins (1995) and Jenkins (1995) have argued, the proportionate contribution of the between-group component (B) to overall inequality is the income inequality literature's analogue of the R 2 statistic used in regression analysis: the size of this contribution is a measure of the amount of inequality that can be 'explained' by the factor (or factors) used to subdivide the sample which, in this case, is country.

Only inequality indices which belong to the family of Generalised Entropy Indices are additively decomposable (Shorrocks (1980)). These indices are defined by a parameter and, when =0, the weights are the population shares of the different groups (that is, /

k k w N N = )
; since the weights sum to unity, the within-group contribution A is a weighted average of the inequality levels within the groups. When =0, the inequality index takes the form: The value of the MLD index, for the 276,150 students living in 41 different countries, was 0.024 and the within-country contribution (the value of A, above) to this value was 0.017 -thus, 28 percent of inequality in the distribution of the S i between the students was due to differences between the countries in the mean values of their scores and 72 percent was due to inequality within the countries. Another division of students which we analysed was the highest occupational class of a parent: white collar, high skilled (49 percent of students); white collar, low skilled (22 percent of students); blue collar, high skilled (15 percent of students); blue collar, low skilled (14 percent of students). The value of the MLD index, for the 266,329 students whose parental occupational status was recorded, was 0.024 and the within-country contribution (the value of A, above) to this value was 0.021 -thus, 13 percent of inequality in the distribution of the S i between the students was due to differences in the mean values of scores between the occupational classes and 87 percent was due to inequality in student scores within the classes.

Another partition we examined was by the level of parental education. The PISA data also provided information on the level of parental education in terms with the highest level of educational attainment of a parent converted into "years of schooling" -the values of this variable ranged from 0, in steps of one year, to 17.

From information on parental "years of schooling" we classified students according to whether their parents' educational level was: low (0-6 years, corresponding to the lowest decile of "years of schooling"); medium (7-15 years, corresponding to 10 th -75 th percentile); and high (>15 years, corresponding to the top quartile). Of the 267,511 students whose parental "years of schooling" was recorded, 11 percent had parents with "low" education; 66 percent had parents with "medium" education; and 23 percent had parents with "high" education.

The value of the MLD index, for these 267,511 students whose parental educational level was recorded, was 0.024 and the within-country contribution (the value of A, above) to this value was 0.022 -thus, 9 percent of inequality in the distribution of the S i between the students was due to differences between the mean This table shows that Tunisia (52 percent), Brazil (51 percent), Indonesia (50 percent), Uruguay (30 percent), Mexico (26 percent), Turkey (26 percent), and Serbia (18 percent) had the largest proportion of students at the lowest level of proficiency in mathematics. In all the other countries, less than one in ten students -and, in many countries, less than one in twenty students -were at the lowest proficiency level. At the other extreme, Hong Kong (10 percent), Belgium (8 percent), Japan, Korea, and the Netherlands (all 7 percent) had the largest proportion of students at the highest level of proficiency in mathematics. From these seven levels of proficiency, we conducted a more detailed analysis of students who were "mathematically weak" and those who were "mathematically strong".

Mathematical Proficiency

Suppose that a student ( i ) is regarded as mathematically weak if his/her SAS in mathematics is lower than (or equal to) some threshold value represented by z, the "weakness line", and as mathematically strong if his/her SAS in mathematics is greater than (or equal to) some threshold value represented by w, the "strength line".

Then, an aggregate measure of mathematical weakness is provided by any of the , where N and M are, respectively, the total number of students and the total number of mathematically weak students.

2.

The Weakness Gap Ratio (P) is the mean distance of the SAS of mathematically weak students from the weakness line, expressed as a proportion of the weakness line:

1 ( ) 1 M W i i z X P Mz z µ = = =
, where: µ W is the mean score of the mathematically weak students..

3.

The Achievement Gap Ratio ( Q) is the mean distance of the outcomes of all the students from the weakness line -the non-weak being assigned a distance of zero -expressed as a proportion of the deprivation line:

1 1 ( ) ( ) M M i i i i z X z X M Q P H Nz Mz N = = = = = × 4.
The Sen (1976) Measure:

(1 ) weakness index in the sense that, in addition to taking account of the proportion of mathematically weak students and the depth of their weakness, it also takes account of inequality in scores between mathematically weak students or, as Sen (1976) would have termed it, "relative weakness".

In a manner analogous to that described above we can define aggregate measures of mathematical strength: the head count ratio; the strength gap ratio; the achievement gap ratio; and the Sen index. Tables 3 and4 show the values of the mathematical weakness and mathematical strength rates for each of the 41 countries in the sample, where a student was regarded as "mathematically weak" if he/she was placed at the lowest level of proficiency (level 1) and as "mathematically strong" if he/she was placed at the highest two proficiency levels (levels 6 and 7).

Table 3 shows, under the column "head count ratio", that 12.2 percent of all students were mathematically weak, their scores placing them in the lowest proficiency level; the "Weakness Gap Ratio" shows that the average score of these weak students was 13 percent below the level 1 threshold score of 357.77; the "Achievement Gap Ratio" shows that the mean distance of all students (weak and non-weak) from the level 1 threshold was 1.6 percent; lastly, the 12 Index. However, there are some exceptions to this rule: Mexico, Turkey, and Serbia have high head count ratios but the mean scores of their weak students are not far below the level 1 threshold and, further, in these countries, there is not much inequality in the scores of their mathematically weak students: consequently, in spite of having a high head count ratio, these countries have a low value on the Sen Index.

Table 4 shows, under the column "head count ratio", that 11.8 percent of all students were mathematically strong, their scores placing them in the two highest proficiency levels, levels 6 and 7; the "Strength Gap Ratio" shows that the average score of these weak students was 6.9 percent above the level 6 threshold score of 606.99; the "Achievement Gap Ratio" shows that the mean distance of all students (strong and not-strong) from the level 6 threshold was 0.82 percent; lastly, the Sen ratio bumps up the Achievement Gap Ratio by the amount of inequality between strong students, as measured by the Gini coefficient, to arrive at a figure of 1.2 percent. Mathematical strength, as measured by the Sen Index, was greatest (2 percent or higher) Australia, Belgium, Czech Republic, Hong Kong, Japan, Korea, Netherlands, and New Zealand and least (less than 0.5 percent) in inter alia Brazil, Indonesia, Mexico, Portugal, Thailand, Tunisia, Uruguay.

Explaining Mathematical Proficiency

Table 5 details the results of a regression which seeks to explaining the predicted Scores in Mathematics by using the socio-economic characteristics of the student and their family along with information about their education (both inside and outside the school environment) and a county-type control variable.

The socio-economic characteristic explanatory variables for the student and the family comprise

• The age of the student: we would expect an older student to perform better in tests, ceteris paribus; however, it is worth pointing out that given that the students in the survey are fairly tightly clustered in terms of age with a standard deviation for age in the full sample of 3½ months so we may not find this to be statistically significant.

• The type of family: single parent, a nuclear family (mother, father and children), mixed family or other. The traditional view would be that students who are members of a nuclear family would perform better in education.

• The level of parental education converted into years of schooling. Here we would expect that more schooling for the parent should improve the educational performance of the student..

• Whether the language used by the family at home is the same as the language used for the test. It is plausible that those students using a language at home other than the test language would under-perform in a test conducted in the test language but it is also possible that such students may have a predisposition to working harder to overcome this disadvantage which will outlay the expectation of underperformance. should perform better than students whose parents who are blue collar workers (i.e. manual). Within these two groups, students with parents who are high skilled workers should perform better than students with parents who are low skilled workers.

The information about their education (both inside and outside the school environment) comprise:

• The minutes of mathematical instruction at School, categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score would rise as the amount of instruction increased.

• The interest in Mathematics, categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score to rise as the interest in Mathematics increased.

• The availability of computing resources at home categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score rises as the availability of computing resources increases; this is both due to computers being of use for improving educational attainment and the presence of computers being a partial proxy for higher household income.

• The availability of other resources at home such as a quiet place to study, categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would • The level of motivation, categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score to rise as the level of motivation increased.

• The level of discipline in the classroom, categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score to rise as the level of discipline in the classroom increased.

• The use of different learning strategies. Here we are comparing the use of elaboration learning strategies (which is the residual category -an example being to understand new concepts in mathematics by relating them to things already known) with memorisation learning strategies (example: learn the answers to problems off by heart) and control learning strategies (example: self testing as the students studies to see if they remember the work already done). Our a priori belief was that elaboration and control learning strategies which were more conducive to a higher test score compared to memorisation strategies.

Focusing on the statistically significant results, our main findings were:

• A student from a nuclear family or a mixed family was predicted to score approximately 6 more points than a student from a single parent family. • Students whose parents were high skilled white collar workers were predicted to score 31 more points than students whose parents were low skilled blue collar workers; students whose parents were low skilled white collar workers were predicted to score 17 more points than students whose parents were low skilled blue collar workers; students whose parents were high skilled blue collar workers were predicted to score 6 more points than students whose parents were low skilled blue collar workers.

• Students for whom the language used by the family at home was different from the language used for the test scored 11 points less than students for whom the language used by the family at home was the same as the language used for the test scored.

• Students in the lowest 2 quartiles for minutes of mathematical instruction at School scored approximately 12 points less than students in the highest quartile; students in the 3 rd quartile scored approximately 7 points fewer than students in the highest quartile. • Students in the lowest quartile for their level of motivation scored approximately 2 points fewer than students in the highest quartile as did students in the third quartile (the coefficient for the second quartile was not statistically significantly different from zero)

• Students in the lowest quartile for the level of discipline in the classroom scored 31 fewer points than students in the highest quartile; students in the second quartile for the level of discipline in the classroom scored approximately 22 fewer points than students in the highest quartile; students in the third quartile for the level of discipline in the classroom scored approximately 11 fewer points than students in the highest quartile.

• Students mostly making use of a memorisation learning strategy scored 2 points less than those mostly making use of elaboration learning strategies;

Students mostly making use of a control learning strategy scored 5 points less than those mostly making use of elaboration learning strategies. • Students in english-speaking OECD countries scored 19 points more than students in OECD partner countries; students in non-English speaking OECD countries scored 18 points more than students in OECD partner countries;

• Students in a higher grade (or year) score 27 points more for each grade.

• The high level of significance for the constant term suggests that there are some omitted variables from the analysis, but this was always likely to be the case when analysing such a complex area.

Conclusion

Looking at the aggregate measures of mathematical strength, the value of the Sen index for mathematical weakness -remembering that this index took into account the proportion of students who were weak, the depth of their weakness, and inequality between mathematically weak students -was lowest (below 0.5%) in Finland, Korea, Netherlands, Canada, The Czech Republic, Macao and Ireland and highest (above 5%) in Uruguay, Indonesia, Tunisia and Brazil. At the other end of the scale, the value of the Sen index for mathematical strength -remembering that this index took into account the proportion of students who were strong, the height of their strength, and inequality between mathematically strong students -was highest in Australia, Belgium, Czech Republic, Hong Kong, Japan, Korea, Netherlands, and New Zealand and least (less than 0.5 percent) in inter alia Brazil, Indonesia, Mexico, Portugal, Thailand, Tunisia, Uruguay.

In terms of the determinants of the point score, the strongest influences (defined as effecting an at least 10 point increase in the point score) on mathematical performance -and in which where policy could play little or no role -were social class (where students with parents who were high skilled white collar workers being predicted to score 31 points more than students whose parents were low skilled blue collar workers) and students being non-native language speakers (with students, whose language in the family home was different to the language used for the test, scoring 11 points less than students whose home language was also the test language)

. The areas in which policy could have an effect were the amount of There is, therefore, evidence from this analysis of the PISA data that if policy makers wish to improve the level of mathematical ability of their school pupils, then a sensible policy regime would be to: (i) increase the amount of mathematical instruction at school, (ii) increase the availability of computers at home (such as laptop borrowing schemes), (iii) increase the availability of other educational resources at home (such as by encouraging the borrowing of library books, CD-ROMs, DVD-ROMs and other educational material), (iv) take measures to develop an higher level of interest in Mathematics, (v) take measures to maintain an higher level of discipline in the classroom.

F

Introduction

An interesting -but, arguably, not sufficiently appreciated -feature of the labour market is the positive association between a person's earnings and whether he or she had studied mathematics at school to an advanced level (Kenny et. al., 1979, Dolton and Vignoles, 2000, McIntosh and Vignoles, 2000). Dolton & Vignoles (2000, 2002) measured the returns an individual obtained from having an A-level in Mathematics (a Mathematics qualification taken in British schools at post-16 level) and found that the return became evident at a later, rather than earlier, stage of a person's career; they argued that the return was due to employers observing that employees with A-level mathematics had higher levels of productivity compared to those who did not have this qualification. They concluded that individuals with an A-level in mathematics earned 7%-10% more than similarly educated workers without this qualification, even after controlling for the initial ability of these individuals.

In a similar vein, Jenkins et al (2003) found that, for women, mathematical ability was particularly important in determining which of them would undertake lifelong learning that would lead to a qualification -an important consideration given the emphasis on lifelong leaning in recent UK government policy thinking such as the "new skills agenda". More recently, [START_REF] Kounine | The Value of Mathematics[END_REF] have bemoaned the decline of mathematics in the United Kingdom and argued that winning the battle of the "maths economy" will be crucial to the UK's future economic success. 1 Murnane et al (1995) and Ingram & Neumann (2006) while supporting this claim, also presented evidence that, in the United States, mathematical qualifications had become increasingly important in determining wage rates: Ingram & Neumann (2006) 1 For a contrary view on the usefulness of mathematics see [START_REF] Jenkins | Maths? I breakfasted on quadratic equations, but it was a waste of time[END_REF]. What benefits the individual also benefits the economy: at a macroeconomic level, studies making use of international school test score data have found a link between mathematical ability and the economic growth of the state (Hanushek and Kimko, 2000). This linkage has been recognised by policy makers. For example, Alan

Greenspan giving testimony before the Committee on Education and the Workforce of the U.S. House of Representatives in March 2004 noted that:

"Research on wealth creation in both emerging and developed nations strongly suggests that it is the knowledge and the skill of our population interacting under our rule of law that determine our real incomes… A study conducted in 1995 revealed that, although our fourth-grade students were above average in both math and science, by the time they reached their last year of high school they had fallen well below the international average." (Greenspan, 2004, emphasis added).

Given the importance of the level of mathematical ability of a country's population, in determining both individual life chances and also macroeconomic performance, this paper asks two broad questions:

(i) How do levels of mathematical ability differ between countries and are levels of inter-country inequality in mathematical ability susceptible to analysis using the tools of inequality theory?

(ii) What are the factors that influence such ability and, in exercising such influence, what is the relative strength of the relevant factors?

We answer these questions using data from the OCED's Programme for International Student Assessment (PISA) which is one of a range of trans-national The issue of inter-country differences in education achievement has been investigated by Maas and Criel (1982) who estimated Gini coefficients based on enrolment data for 16 East African countries and, more recently, by [START_REF] Thomas | Measuring Education Inequality: Gini Coefficients of Education[END_REF][START_REF] Thomas | A New Dataset on Inequality in Education: Gini and Theil Indices of Schooling for 140 countries, 1960-2000[END_REF] who studied inequality in educational attainments for 140 countries. The latter set of papers developed the concept of education Gini and argued that this could be used as an indicator of welfare complementing average educational, health, nutritional, and income attainments. In so doing, [START_REF] Thomas | Measuring Education Inequality: Gini Coefficients of Education[END_REF][START_REF] Thomas | A New Dataset on Inequality in Education: Gini and Theil Indices of Schooling for 140 countries, 1960-2000[END_REF] were motivated by [START_REF] Sen | Economic Policy and Equity: An Overview[END_REF] observation that concern with equity should not be confined to just income inequality but, indeed, should be extended to embrace all the dimensions which impinge on a person's ability to function effectively in society. born in 1958 and 1970, they found that the links between educational achievement and parental income / social class strengthened over this period so that the labour market success or failure of individuals became more closely connected to their parents income. This work is complemented by that of [START_REF] Galindo-Rueda | The Widening Gap in UK Higher Education[END_REF] who found that in UK higher education, even before the introduction of tuition fees, children from poorer neighbourhoods had become less likely to participate in higher education since 1994-95, as compared to children from richer neighbourhoods.

In this paper we extend the contribution of [START_REF] Thomas | Measuring Education Inequality: Gini Coefficients of Education[END_REF][START_REF] Thomas | A New Dataset on Inequality in Education: Gini and Theil Indices of Schooling for 140 countries, 1960-2000[END_REF], to the study of educational inequality, in one respect. Following the work of Anand and Sen (1997) we argue that the average achievements of a country with respect to a particular welfare indicator -which in the case of this paper is "mathematical ability"

as measured by PISA -should be tempered by considerations of inequality in the distribution of achievements between the individuals in the country's population. Anand and Sen (1997) referred to the resultant indicators of achievement as being "equity sensitive". In this paper we construct, for each country in the PISA sample, equity sensitive indicators of mathematical ability.

The heart of the paper lies, however, in answering the latter questions relating to the factors which enhance mathematical ability. Since, as noted above, data from PISA contained a wealth of information on the circumstances of the students, particularly in terms of their home and school environment, we identify, and examine the relative influence of, factors which serve to enhance the mathematical performance of students in the PISA assessment. Jenkins et. al. (1993) examined the determinants of lifelong learning and concluded that those who left school with O-level qualifications or higher were much more likely to undertake lifelong learning. [START_REF] Okpala | Student Factors as Correlates of Acievement in Physics[END_REF]

Equity Sensitive Indicators of Student Assessment Scores

In a paper prepared for the 1995 Human Development Report, Anand and Sen (1997) pointed out that a country's non-economic achievements were likely to be unequally distributed between subgroups of its population: for example, in terms of gender equality, which was the focus of their concern, the female literacy rate, or female life expectancy, was often lower than that for males. In the face of such intergroup inequality, they argued that a country's achievement with respect to a particular outcome should not be judged exclusively by its mean level of achievement (for example, by the average literacy rate for a country) but rather by the mean level adjusted to take account of inter-group differences in achievements. Anand and Sen (1997) proposed a method, based on Atkinson's (1970) seminal work on the relation between social welfare and inequality, for making such adjustments and they termed the resulting indicators equity sensitive indicators. They further suggested that assessments of country achievements should be made on the basis of such equity sensitive indicators rather than, as was often the case, on the basis of its mean level of achievement. This would, then, allow a comparison between two countries, one of which had a lower mean achievement level, but a more equitable distribution of achievement, than the other. 3 In this section we apply these ideas to the student assessment scores (SAS) -hereafter, simply, "scores" -in the PISA data.

Suppose that X is the average score in a country where i X is the score of student i (i=1…N). We know that, because of inequality in the distribution of scores between students, the average score of a country will not be achieved by all its students.

3 Anand and Sen (1997) compared the Honduras (with an average literacy rate of 75%, distributed between men and women as 78%, 73%) with China (with an average literacy rate of 80%, distributed between men and women as 92%, 68%) and asked which country should be regarded as having the "better" achievement with regard to literacy: China with a higher overall rate or the Honduras with greater gender equality? The answer to this question depends on how averse we are to inequality. In his seminal paper on income inequality, Atkinson (1970) argued that we (society) would be prepared to accept a reduction in average income, provided the lower income was equally distributed, from a higher average income which was unequally distributed. 4The size of this reduction depended upon our degree of "inequality aversion" which Atkinson (1970) measured by the value of a (inequality aversion) parameter, 0 .

When 0 = , we are not at all averse to inequality implying that we would not be prepared to accept even the smallest reduction in average income in order to secure an equitable distribution. The degree of inequality aversion increases with the value of : the higher the value of , the more averse we would be to inequality and, in order to secure an equitable distribution of income, the greater the reduction in average income we would find acceptable.

These ideas can, equally well, be applied to student assessment scores. We can reduce the average score, X , of a country by the amount of inter-student inequality in scores to arrive at e X , a "group equity sensitive" score for the country, e X X . We refer to e X as the equally distributed equivalent score. The size of this reduction (as given by the difference, e X X ) depends upon our aversion to inequality: the lower our aversion to inequality, the smaller will be the difference and, in the extreme case in which there is no aversion to inequality ( 0 = ), there will be no difference between the average and the equity sensitive scores. Three special cases, contingent upon the value assumed by , may be distinguished:

X X = 2. When 1 = , e X
is the geometric mean of the student scores:

1/N 1 < N e i i X X X = = .
3. When 2 = , e X is the harmonic means of the student scores:

( )

1 1 e N i i N X X X = = < .
Table 1 shows the equity sensitive scores for each of the 41 countries in the PISA data, contingent upon the amount of inequality in the distribution of scores between all the 15 year olds in (the sample for) that country. This table shows three separate rankings for the 41 countries: first, the ranking that resulted entirely from the average score of countries when intra-country inequality was deemed not to matter; second, the ranking that resulted when inequality aversion was measured by 1 = or, in other words, the relevant average was measured by the geometric mean; lastly, the ranking that resulted when inequality aversion was measured by 2 = or, in other words, the relevant average was measured by the harmonic mean.

Table 1 shows that the rankings changed only slightly from no inequality aversion to inequality aversion: the first 8 places were unchanged; Australia slipped from 11 th to 12 th place while Canada rose from 12 th to 11 th place; Germany slipped from 19 th to 20 th to 21 st under successively higher degrees of inequality aversion while Italy slipped from 24 th to 25 th place. In general, we can conclude that the use of equity-sensitive indicators did not add much to the rankings of countries by their average level of mathematical achievements. Table 2 shows that Tunisia and Brazil (52 percent), Indonesia (51 percent), Uruguay (30 percent), Mexico (26 percent), Turkey (26 percent), Thailand (18 percent), and Greece and Serbia (18 percent) had the largest proportion of students at the lowest level of proficiency in mathematics. In all the other countries, less than one in ten students -and, in many countries, less than one in twenty students -were at the lowest proficiency level.

F

At the other extreme, Table 3 shows that Hong Kong (32 percent),

Netherlands and Belgium (27 percent), Liechtenstein and the Czech Republic (25 percent), and Japan and Korea (24 percent) had the largest proportion of students at the two highest levels of mathematical proficiency.

Table 4 details the results of a regression model which seeks to explaining the predicted Scores in Mathematics by using the socio-economic characteristics of the student and their family along with information about their education (both inside and outside the school environment) and a county-type control variable.

The socio-economic characteristic explanatory variables for the student and the family comprise

• The age of the student: we would expect an older student to perform better in tests, ceteris paribus; however, it is worth pointing out that given that the students in the survey are fairly tightly clustered in terms of age with a 10 standard deviation for age in the full sample of 3½ months so we may not find this to be statistically significant.

• The type of family: single parent, a nuclear family (mother, father and children), mixed family or other. The traditional view would be that students who are members of a nuclear family would perform better in education.

• The level of parental education converted into years of schooling. Here we would expect that more schooling for the parent should improve the educational performance of the student..

• Whether the language used by the family at home is the same as the language used for the test. It is plausible that those students using a language at home other than the test language would under-perform in a test conducted in the test language but it is also possible that such students may have a predisposition to working harder to overcome this disadvantage which will outlay the expectation of underperformance.

• Students with parents who are white collar workers (i.e. non-manual) should perform better than students whose parents who are blue collar workers (i.e. manual). Within these two groups, students with parents who are high skilled workers should perform better than students with parents who are low skilled workers. This would be consistent with results on socio-economic educational inequality reported by [START_REF] Machin | Educational inequality: the widening socio-economic gap[END_REF]Vignoles (2004) and[START_REF] Galindo-Rueda | The Widening Gap in UK Higher Education[END_REF].

The information about their education (both inside and outside the school environment) comprised the following items: (ii) The interest in Mathematics: these were categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score to rise as the interest in Mathematics increased.

(iii)The availability of computing resources at home: these were categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score rises as the availability of computing resources increases; this is both due to computers being of use for improving educational attainment and the presence of computers being a partial proxy for higher household income.

(iv)The availability of other resources at home (such as a quiet place to study):

these were categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category.

We would expect the test score to rise as the availability of other resources at home increased.

(v) The level of motivation: this was categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score to rise as the level of motivation increased. With this background, we use the regression estimates shown in Table 4 to highlight the main findings.

Country Effects. One would expect that, after controlling for other variables, the country in which a student lived would influence his/her results. This is because the country of residence would capture the general level of resources available to residents and which would buttress the more specific variables pertaining to the individual students. Ideally, the equations should have been estimated with a dummy for each country. However, given that there were 41 countries, we decided to group the countries as follows:

(i) OECD English speaking countries (Australia, Canada, Ireland, New Zealand, UK, and USA).

(ii) OECD Nordic countries (Denmark, Finland, Iceland, Norway, Sweden). Family Type and Parental Occupation. The results showed very clearly that both family structure and parental occupation exercised a significant influence on student performance: ceteris paribus a student from a nuclear family was predicted to score approximately 6 more points, on average, than a student from a single parent family while students whose parents were high skilled white collar workers were predicted to score an average of 29 more points than students whose parents were low skilled blue collar workers.

F
This parental advantage persisted for students whose parents were lower down the occupational ladder: students whose parents were low skilled white collar workers were predicted to score 12 more points than students whose parents were low skilled blue collar workers while students whose parents were high skilled 5 Meaning that, ceteris paribus, their average scores were higher than the OCED partner average by the coefficient estimate. Both these results are consistent with received wisdom: [START_REF] Machin | Educational inequality: the widening socio-economic gap[END_REF] have drawn attention to the links between educational achievement and parental class; [START_REF] Gordon | Family Structure, Educational Achievement, and the Inner City[END_REF] has argued that the current rates of success in the GCSE exams in Britain (taken at the age of 16) are associated with locations away from inner cities and the reason for this is the high concentration of lone parent families in inner cities.

Years of Schooling and Language at Home. The effect on scores of how well educated students' parents were was significant, but weak. A student, one of whose parents had received 17 years of parental schooling (approximately equivalent to a Master's degree) was predicted to score approximately only two more points than a student with 10 years of parental schooling (approximately equivalent to leaving at age 16). This suggests that, once parental occupational class was controlled for, there was not much additional role for the influence of parental educational.

Similarly, the language spoken at home did exert a large and significant effect on student scores: students, for whom the language used by the family at home was different from the language used for the test, scored an average of 16 fewer points compared to students for whom the language used by the family at home was the same as the language used for the test.

The School Environment. The environment at school affected student performance in a number of ways:

1. First, and most obvious, was the time which the school devoted to instruction in mathematics. There was a very clear correlation between student 4. Fourth, the school had an influence on student scores through the choice of appropriate teaching and learning strategies. However, it must be admitted that after controlling for the other school factors, 1-3 above, the influence of learning strategies on student performance was weak.

The results relating to school environment have obvious implications for the efficiency with which schools are run. In this connection, [START_REF] Bee | Costs and Economies of Scale in UK Private Schools[END_REF] offer evidence that there may be economies of scale to be reaped from having larger schools. 7

The Home Environment: The results clearly pointed to the importance of computing resources at home as an aid to proficiency in mathematics. Students in the lowest quartile for the availability of computing resources at home scored, on average, approximately 47 points fewer than students in the highest quartile while students in the second quartile for the availability of computing resources at home scored approximately 23 points less than students in the highest quartile.

Interestingly, there was hardly any difference in the average scores of students in the third and fourth quartiles of computing facilities thus suggesting that the marginal benefit of better facilities diminished very rapidly.

Non-computing resources -for example, in the form a quiet, separate place to study -were also important in influencing student scores. Students in the lowest quartile for the availability of other resources at home scored approximately 31

7 See also Dolton (1991) for a study of the efficient provision of compting services in UK universities.

Conclusion

The proportion of mathematically weak students was lowest (3 percent or less) in Finland, Korea, the Netherlands, Macao, Canada, the Czech Republic, and Hong Kong and highest (over 50 percent) in Tunisia, Brazil, and Indonesia. At the other end of the scale, the proportion of mathematically strong students was highest (25 percent or more) in Hong Kong, the Netherlands, Belgium, Liechtenstein, and the Czech Republic.

In terms of the determinants of the point score, the strongest influences (defined as effecting, at least, a10 point increase in the point score) on mathematical performance -and in which where policy could play little or no role -were social class) and students being non-native language speakers .The areas in which schooling policy could have an effect were the amount of mathematical instruction at school and the level of classroom discipline. In terms of the home environment, the availability of a computer was important but equally important was the availability of noncomputing facilities (quiet place to study, books etc.). Indeed, as the results showed the marginal benefit of computing facilities diminished much more rapidly than the marginal benefit from non-computing facilities.

A very important factor for high mathematical achievement is an interest in mathematics and here a number of factors -employment opportunities, school, home, friends -need to coalesce to create, sustain, and enhance interest in mathematics.

There is, therefore, evidence from this analysis of the PISA data that if policy makers wish to improve the level of mathematical ability of their 15 year olds, then a sensible policy regime would be to: (i) increase the amount of mathematical instruction at school and classroom discipline in general, (ii) increase the availability of computers at home (such as laptop borrowing schemes), (iii) increase the

  score over the entire sample of students. This inequality index isTheil's (1967) Mean Logarithmic Deviation (MLD) index and, because of its attractive features in terms of the interpretation of the weights, it was the one we used to decompose inter-student inequality in scores by various grouping of students.

  the different parental educational levels and 91 percent was due to inequality in student scores within the parental educational levels.

  G W is the Gini coefficient computed over the scores of mathematically weak students 3 . If there was no inequality between mathematically weak students in respect of their SAS (i.e. 0 W G = ) then S H P Q = × = . But, if there was inequality between mathematically weak students in their SAS (i

•

  Students with parents who are white collar workers (i.e. non-manual)

  score to rise as the availability of other resources at home increasesd.

  ability. 2 PISA is a collaborative effort, involving all OECD countries and a significant number of partner countries, to measure how well 15 year students "are prepared to meet the challenges of today's knowledge society". The PISA 2003 assessments consist of paper-and-pencil tests and the following domains are tested: (i) mathematical literacy; (ii) reading literacy; (iii) scientific literacy.On the basis of these tests, each of 276,150 students in 41 countries was assigned a score for mathematics, reading, and science. In addition to this information about how well students fared in their assessments, the PISA data contains a wealth of information on the circumstances surrounding a student. These relate to inter alia: (i) his/her personal circumstances, living arrangements etc; (ii) parental attributes relating to education, class; (iii) home possessions and environment relating to books, computers, internet, place to study; (iv) school circumstances relating to amount of instruction, relationship with teachers, type of school etc.

  (2004) have studied educational inequality in the UK from the perspective of socio-economic groups. Comparing two cohorts of individuals,

  the factors underlying educational achievement have been the subject of several studies.

  assessing the SAS achievement of a country, by how much should we reduce its average SAS to take account of inequality in scores?

  inequality aversion), e X is the arithmetic mean of the student scores: e

  The minutes of mathematical instruction at School: these were categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score would rise as the amount of instruction increased.

  The level of discipline in the classroom, categorised by us into four quartiles ranging from lowest to highest, with the highest quartile taken as the residual or base category. We would expect that the test score to rise as the level of discipline in the classroom increased.(vii) The use of different learning strategies. Here we are comparing the use of elaboration learning strategies (which is the residual category -an example being to understand new concepts in mathematics by relating them to things already known) with memorisation learning strategies (example: learn the answers to problems off by heart) and control learning strategies (example: self testing as the students study, to see if they remember the work already done). Our a priori belief was that elaboration and control learning strategies were more conducive to a higher test score compared to memorisation strategies.

  were predicted to score 2 more points than students whose parents were low skilled blue collar workers.

  students in the highest quartile; students in the second quartile for the availability of other resources at home scored approximately 15 fewer points than students in the highest quartile. With respect to non-computing resources, there was a 10 point difference in the average scores of students in the third and fourth quartiles of non-computing facilities suggesting that the marginal benefit of better non-computing facilities did not diminish as rapidly as did the marginal benefit of better computing facilities.

  educational resources at home (such as by encouraging the borrowing of library books, CD-ROMs, DVD-ROMs and other educational material), (iv) take measures to develop an higher level of interest in Mathematics.

Table 1 Equity Sensitive Assessment Scores in Mathematics of 15 year-olds, by Country

 1 Figures in parentheses are values of the Gini Coefficient.

	o r
	P
	e
	e r
	R e v
	i e
	w

Table 2 Proficiency Levels in Mathematics of 15 year-olds, by Country

 2 The numbers in the line against the country name represent student numbers at each proficiency level. The numbers in italics, below that line, represent student percentages at each proficiency level.

	Country				Level				Total
		1	2	3	4	5	6	7
	1.Australia	551	1,300	2,360	3,518	2,401	1,772	649	12,551
		4.39	10.36	18.80	28.03	19.13	14.12	5.17	100.00
	2. Austria	190	567	956	1,358	859	509	158	4,597
		4.13	12.33	20.80	29.54	18.69	11.07	3.44	100.00
	3. Belgium	537	769	1,366	2,127	1,632	1,657	708	8,796
		6.11	8.74	15.53	24.18	18.55	18.84	8.05	100.00
	4. Brazil	2,289	1,058	662	318	81	33	7	4,448
		51.46	23.79	14.88	7.15	1.82	0.74	0.16	100.00
	5. Canada	749	2,539	5,869	8,827	5,504	3,483	982	27,953
		2.68	9.08	21.00	31.58	19.69	12.46	3.51	100.00
	6. Czech Rep	180	606	1,161	1,634	1,151	1,076	512	6,320
		2.85	9.59	18.37	25.85	18.21	17.03	8.10	100.00
	7. Denmark	162	470	872	1,324	775	472	143	4,218
		3.84	11.14	20.67	31.39	18.37	11.19	3.39	100.00
	8. Finland	68	287	942	1,932	1,276	970	321	5,796
		1.17	4.95	16.25	33.33	22.02	16.74	5.54	100.00
	9. France	188	443	865	1,336	819	528	121	4,300
		4.37	10.30	20.12	31.07	19.05	12.28	2.81	100.00
	10. Germany	344	556	899	1,244	866	591	160	4,660
		7.38	11.93	19.29	26.70	18.58	12.68	3.43	100.00
	11. Greece	850	1,016	1,280	956	381	125	19	4,627
		18.37	21.96	27.66	20.66	8.23	2.70	0.41	100.00
	12. Hong Kong	133	251	582	1,059	1,004	1,007	442	4,478
		2.97	5.61	13.00	23.65	22.42	22.49	9.87	100.00
	13. Hungary	339	739	1,196	1,342	689	362	98	4,765
		7.11	15.51	25.10	28.16	14.46	7.60	2.06	100.00
	14. Iceland	132	340	703	1,033	649	398	95	3,350
		3.94	10.15	20.99	30.84	19.37	11.88	2.84	100.00
	15. Indonesia	5,431	3,169	1,537	516	96	12	0	10,761
		50.47	29.45	14.28	4.80	0.89	0.11	0.00	100.00
	16. Ireland	147	436	936	1,269	673	351	68	3,880
		3.79	11.24	24.12	32.71	17.35	9.05	1.75	100.00
	17. Italy	788	1,558	2,638	3,616	1,798	988	253	11,639
		6.77	13.39	22.67	31.07	15.45	8.49	2.17	100.00
	18. Japan	208	391	779	1,275	941	783	330	4,707
		4.42	8.31	16.55	27.09	19.99	16.63	7.01	100.00
	19. Korea	119	368	949	1,554	1,157	913	382	5,442
		2.19	6.76	17.44	28.56	21.26	16.78	7.02	100.00
	20. Latvia	297	692	1,249	1,409	626	288	66	4,627
		6.42	14.96	26.99	30.45	13.53	6.22	1.43	100.00
	21. Liechtenstein	14	22	60	83	69	63	21	332
		4.22	6.63	18.07	25.00	20.78	18.98	6.33	100.00
	22. Luxembourg	259	570	910	1,193	609	308	74	3,923
		6.60	14.53	23.20	30.41	15.52	7.85	1.89	100.00
	23. Macao	32	110	266	402	225	173	42	1,250
		2.56	8.80	21.28	32.16	18.00	13.84	3.36	100.00
	24. Mexico	7,899	9,563	8,020	3,781	611	105	4	29,983
		26.34	31.89	26.75	12.61	2.04	0.35	0.01	100.00
	25. Netherlands	100	291	677	1,069	761	827	267	3,992
		2.51	7.29	16.96	26.78	19.06	20.72	6.69	100.00
	26. New Zealand	188	448	857	1,234	849	654	281	4,511

Table 3 Rates of Mathematical Weakness of 15-year olds, by Country *

 3 Students at the lowest level of mathematical proficiency.

	Country	Head Count	Weakness Gap	Achievement	Sen Measure (%)
		Ratio (%)	Ratio (%)	Gap Ratio (%)
	1.Australia	4.4	10.9	0.48	0.70
	2. Austria	4.1	8.9	0.37	0.53
	3. Belgium	6.1	13.0	0.79	1.1
	4.Brazil	51.5	19.1	9.8	13.6
	5. Canada	2.7	8.3	0.22	0.33
	6. Czech Rep	2.8	8.1	0.23	0.33
	7. Denmark	3.8	10.3	0.40	0.58
	8. Finland	1.2	8.1	0.09	0.13
	9. France	4.4	9.3	0.41	0.59
	10. Germany	7.4	10.9	0.80	1.16
	11. Greece	18.4	13.2	2.4	3.5
	12. Hong Kong	3.0	12.2	0.36	0.52
	13. Hungary	7.1	10.7	0.77	1.1
	14. Iceland	3.9	9.7	0.38	0.55
	15. Indonesia	50.5	14.9	7.5	10.6
	16. Ireland	3.8	8.6	0.33	0.48
	17. Italy	12.2	13.0	1.6	2.3
	18. Japan	4.4	11.4	0.5	0.74
	19. Korea	2.2	8.9	0.20	0.28
	20. Latvia	6.4	8.6	0.55	0.81
	21. Liechtenstein	4.2	9.9	0.42	0.62
	22. Luxembourg	6.6	10.0	0.66	0.94
	23. Macao	2.6	9.6	0.25	0.35
	24. Mexico	26.3	12.4	3.3	4.7
	25. Netherlands	2.5	7.6	0.19	0.28
	26. New Zealand	4.2	9.4	0.39	0.56
	27. Norway	5.8	10.2	0.59	0.85
	28. Poland	6.0	10.1	0.62	0.90
	29. Portugal	10.5	10.0	1.1	1.5
	30. Russian Fed	9.9	10.2	1.0	1.5
	31. Slovakia	5.1	10.5	0.54	0.80
	32. Spain	5.6	9.9	0.55	0.80
	33. Sweden	5.4	10.9	0.59	0.89
	34. Switzerland	4.5	9.9	0.44	0.64
	35. Thailand	21.4	10.1	2.2	3.1
	36. Tunisia	52.2	16.0	8.4	11.7
	37. Turkey	25.8	12.5	3.2	4.6
	38. UK	3.9	9.1	0.35	0.52
	39. USA	9.3	10.5	0.99	1.4
	40. Uruguay	30.2	16.7	5.0	7.1
	41. Serbia	17.8	10.7	1.9	2.8
	Total	12.2	13.0	1.6	2.3

*

Table 4 Rates of Mathematical Strength of 15-year olds, by Country *

 4 Students at the highest two levels of mathematical proficiency.

	Country	Head Count	Weakness Gap	Achievement	Sen Measure (%)
		Ratio (%)	Ratio (%)	Gap Ratio (%)
	1.Australia	19.3	7.4	1.4	2.0
	2. Austria	14.5	6.8	0.98	1.4
	3. Belgium	26.9	7.9	2.1	3.0
	4.Brazil	0.90	5.7	0.05	0.07
	5. Canada	16.0	6.7	1.1	1.5
	6. Czech Rep	25.1	8.1	2.05	2.9
	7. Denmark	14.6	6.6	0.96	1.4
	8. Finland	22.3	6.9	1.5	2.2
	9. France	15.1	6.0	0.91	1.3
	10. Germany	16.1	5.7	1.07	1.5
	11. Greece	3.1	5.0	0.12	0.23
	12. Hong Kong	32.4	8.0	2.6	3.7
	13. Hungary	9.7	6.2	0.60	0.86
	14. Iceland	14.7	6.2	0.91	1.3
	15. Indonesia	0.11	0.00	2.5	0.00
	16. Ireland	10.8	5.7	0.62	0.90
	17. Italy	10.7	6.3	0.67	0.97
	18. Japan	23.6	7.9	1.9	2.7
	19. Korea	23.80	7.9	1.9	2.7
	20. Latvia	7.7	5.8	0.45	0.65
	21. Liechtenstein	25.3	7.4	1.9	2.7
	22. Luxembourg	9.7	6.1	0.60	0.86
	23. Macao	17.2	6.4	1.1	1.6
	24. Mexico	0.36	3.5	0.01	0.02
	25. Netherlands	27.4	7.1	1.9	2.7
	26. New Zealand	20.7	7.9	1.6	2.3
	27. Norway	10.7	6.3	0.67	0.96
	28. Poland	8.9	6.4	0.57	0.81
	29. Portugal	4.8	4.9	0.24	0.34
	30. Russian Fed	6.8	6.4	0.44	0.63
	31. Slovakia	12.9	6.5	0.83	1.2
	32. Spain	8.1	5.1	0.42	0.60
	33. Sweden	14.8	6.5	0.96	1.39
	34. Switzerland	17.1	7.2	1.2	1.8
	35. Thailand	2.0	5.2	0.11	0.15
	36. Tunisia	0.11	1.51	0.00	0.00
	37. Turkey	4.7	9.6	0.45	0.64
	38. UK	15.1	6.7	1.0	1.4
	39. USA	8.6	5.9	0.50	0.72
	40. Uruguay	2.6	5.3	0.14	0.20
	41. Serbia	2.0	4.3	0.09	0.12
	Total	11.8	6.9	0.82	1.2

*

Table 5 Regression Estimates for Explaining Predicted Scores in Mathematics

 5

	Age

-specific variables (residual -OECD partner countries):

	OECD country: english speaking	19.160***
		(30.18)
	OECD country: non-english speaking	17.791***
		(33.52)
	Constant	478.165***
		(45.72)
	Observations	186606
	R-squared	0.33

Page 32 of 61 Editorial

	Submitted Manuscript
	found that the return to mathematical and verbal ability nearly doubled between 1980
	and 1998.
	F o r
	P
	e
	e r
	R e v
	i e
	w
	2

Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  examined the factors underlying student achievement in physics in Nigeria and examined the role of inter alia gender, home, interest in physics in shaping achievement in physics.

	F o r
	P
	e
	e r
	R e v
	i e
	w
	6

  , the average score of students with lowest level of interest was 27 points lower but, compared to the average score of students with the highest level of motivation to study mathematics, the average score of students with lowest level of motivation was only 4 points lower.

	performance and the time spent on mathematics. For example, compared to the interest in mathematics
	average score of students in the highest quartile with respect to instruction
	time: ceteris paribus the average score of students in the lowest quartile of
	instruction time was 18 points lower; the average score of students in the
	second quartile of instruction time students was 10 points lower, and the
	average score of students in the third quartile of instruction time was 8 points
	lower.
	F o r 2. Second, our results clearly showed that classroom discipline mattered. The effective learning of mathematics depended not just on the time devoted to is study but also on classroom management which sought to create a good F o r
	P learning environment in the classroom through high standards of discipline. P
	e Ceteris paribus compared to the average score of students who enjoyed the e
	e r highest level of classroom discipline, students in the lowest quartile for the e r
	R e v level of discipline in the classroom scored, on average, 31 fewer points. Nor was this poorer performance confined to the lowest level of discipline: students in the second quartile for the level of discipline in the classroom R e v
	i e w scored approximately 21 fewer points than students in the highest quartile, i e while students in the third quartile for the level of discipline in the classroom scored approximately 11 fewer points (compared to students in the highest w
	quartile).
	3. Third, the school could have been instrumental in making students interested
	in mathematics and motivating them to do well. 6 Of these two factors, once
	interest in mathematics had been controlled for, motivation did not have much
	of a role. Compared to the average score of students with the highest level of
	15 16

Table 1 Equity Sensitive Assessment Scores in Mathematics of 15 year-olds, by Country

 1

	Page 55 of 61		Submitted Manuscript
	Country	Sample	Value of			Rank	Rank	Rank
		Size	inequality			under	under	under
			aversion			=0	=1	=2
			parameter
			=0	=1	=2
	1. Hong Kong	4,478	555.9	547.1	537.2	1	1	1
			(0.095)
	2. Finland	5,796	542.8	536.8	530.4	2	2	2
			(0.083)
	3. Netherlands	3,992	542.1	534.3	525.9	3	3	3
			(0.094)
	4. Korea	5,444	540.7	533	524.8	4	4	4
			(0.093)
	5. Liechtenstein	332	536.5	527.4	517.5	5	5	5
			(0.100)
	6. Czech Rep	6,320	535.0	526.1	516.8	6	6	6
			(0.102)
	7. Japan	4,707	533.5	524	513.5	7	7	7
			(0.102)
	8. Belgium	8,796	533.2	521.9	509.3	8	8	8
			(0.111)
	9. New Zealand	4,511	525.6	516.6	507	9	9	10
			(0.103)
	10. Macao	1,250	522.8	515.6	508.1	10	10	9
			(0.092)
	11.Australia	12,551	522.3	513.3	503.6	11	12	12
			(0.102)
	12. Canada	27,953	521.4	514.3	506.9	12	11	11
			(0.091)
	13. Switzerland	8,420	518.2	509.7	500.6	13	13	13
			(0.100)
	14. Iceland	3,350	515.1	507.4	499.2	14	14	14
			(0.095)
	15. France	4,300	514.7	506.9	498.6	15	15	15
			(0.096)
	16. UK	9,535	514.4	506.5	498.1	16	16	16
			(0.097)
	17. Denmark	4,218	513.6	505.9	497.6	17	17	17
			((0.096)
	18. Austria	4,597	511.9	503.9	495.5	18	18	18
			(0.098)
	19. Germany	4,660	508.4	498.4	487.6	19	20	21
			(0.108)
	20. Sweden	4,624	508.0	499.3	489.8	20	19	20
			(0.101)
	21. Ireland	3,880	504.7	497.8	490.6	21	21	19
			(0.092)
	22. Slovakia	7,346	504.2	495.9	487.1	22	22	22
			(0.100)
	23. Italy	11,639	496.0	487.3	477.7	23	25	25
			(0.102)
	24. Norway	4,064	495.6	487.4	478.8	24	24	24
			(0.101)

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 4 Regression Estimates for Explaining Predicted Scores in Mathematics Family Type (residual-other type):

 4

	Single parent family

of Mathematical Instruction at School (residual - fourth quartile):

	Minutes: lowest quartile	-17.85***
		(32.59)
	Minutes: second quartile	-9.54***
		(14.14)
	Minutes: third quartile	-7.62***
		(13.66)

Computing facilities at home (residual -fourth quartile):

	Interest in mathematics (residual -fourth quartile):
	Interest: lowest quartile			-27.26***
				(49.91)
	Interest: second quartile			-17.28***
				(35.60)
	Interest: third quartile			-8.95***
				(14.52)
	Motivation (residual -fourth quartile):
	Motivation: lowest quartile			-4.13**
				(6.16)
	Motivation: second quartile			-2.19
				(4.17)
	Motivation: third quartile			-2.57***
				(4.20)
	Learning Strategies (residual -elaboration learning F strategies): Memorisation/rehearsal learning strategies o r Control learning strategies	1.01** (2.37) 3.57***
	P Discipline in classroom (residual -fourth quartile):	(7.95)
	Discipline: lowest quartile Discipline: second quartile	e e r		-31.20*** (53.26) -20.74*** (40.89)
	Discipline: third quartile			-10.78***
	R e Country-specific variables (residual -OECD partner Student's grade: v countries):	(18.91) 29.66*** (93.32)
	Facilities: lowest quartile Facilities: second quartile OECD country: english speaking OECD Nordic countries OECD Western Europe	i e w	-46.91*** (66.05) -22.83*** 22.23*** (35.25) 23.12*** (29.18) 28.92***
				(31.05) (48.55)
	Facilities: third quartile OCED Eastern Europe			-1.51** 38.49***
				(3.15) (51.75)
	Low home Educational Resources: OECD Others		-2.92*** -16.19***
				(6.20) (22.49))
	Home Resources (residual -fourth quartile): Intercept		502.18***
	Resources: lowest quartile			-30.86*** (271.93)
	Observations			(38.45) 186612
	Resources: second quartile R-squared (adjusted)			-14.46*** 0.346
				(24.27)
	Resources: third quartile Absolute value of t statistics in parentheses		-9.68***
				(15.95)
		29

* significant at 10%; ** significant at 5%; *** significant at 1%

Page 60 of 61 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	F o r
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	e r
	R e v
	i e
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30 Page 61 of 61 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

In the language of economics, the two situations would yield the same level of social welfare, i.e. be 'welfare equivalent'.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

It is important to emphasise three aspects of Sen's measure: it takes account of the number of mathematically weak students, relative to the total number of students, through H, the headcount ratio; it takes account of the depth of their weakness through P, the weakness gap ratio; it takes account of relative weakness through G W , the Gini coefficient calculated on the scores of mathematically weak students.

See Hansen & Vignoles (2005) andBrown et al (2007) for a discussion of how PISA compares to other assessments.

In the language of economics, the two situations would yield the same level of social welfare, i.e. be 'welfare equivalent'.

Though, in addition to the school, several factors -including, home, friends -could influence interest and motivation.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript