the gender gap in mathematics in french primary...

The gender gap in mathematics in French primary school

Adrian Hille

Sciences Po Paris · Ecole Polytechnique · ENSAE

The gender gap in mathematicsin French primary school

Adrian Hille

Programme doctoral

Economics and Public Policy, PhD track

Mémoire de master

Version révisée du 13 juin 2011

(Version originale du 26 mai 2011)

Directeur du mémoire : Denis Fougère (ENSAE/CREST)

2e membre du jury : Yann Algan (Sciences Po)

Année académique 2010-2011

Date de remise : 27 mai 2011

Date de soutenance : 6 juin 2011

Masters Thesis

The gender gap in mathematics in Frenchprimary school

Adrian Hille

Revised version: June 13th, 2011(Original version: May 26th, 2011)

This paper examines the evolution and the determinants of the gender gapin mathematics in French primary school. Using data from the French PrimarySchool Panel, I perform a detailed Oaxaca-Blinder decomposition with the aimto decompose the gender gap at di!erent age levels into an explainable and twostructural components. The underlying model is based on Todd and Wolpins(2003) cognitive skill production function approach. First, I find that the gendergap arises between 3rd and 6th grade of primary school. Second, the gender gapat the top of the distribution develops earlier than for low-achieving students.Third, in 6th grade, girls are predicted to score better than boys both accordingto their endowment and their returns to school inputs. The advantage of boys isthus unexplainable with observed characteristics. Fourth, I find evidence againstthe widely accepted theory explaining the gender gap with superior spatial skillsamong boys. Girls score particularly well both in spatial skills and geometry.Finally, this paper provides some support for the theory attributing the mathe-matics gender gap to stereotypical thinking.

Acknowledgements

I wish to thank Maxime To and Denis Fougere for their help and comments.

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Contents

1 Introduction 3

2 Genes or stereotypes? – Possible explanations for the gender gap in math-ematics 7

3 The econometric model 103.1 A economic model derived from Todd and Wolpin . . . . . . . . . . . . . . . 103.2 The detailed Oaxaca-Blinder decomposition . . . . . . . . . . . . . . . . . . . 153.3 The empirical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 The data 254.1 Standardized tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Non-cognitive skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Background characteristics, family and school input . . . . . . . . . . . . . . 31

5 The gender gap 365.1 The gender gap at the entry into 1st grade . . . . . . . . . . . . . . . . . . . 365.2 At the entry into 3rd grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.3 At the entry into 6th grade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Results of the decomposition 466.1 Evaluation at the entry into primary school (CP) . . . . . . . . . . . . . . . . 466.2 Evaluation at the entry into 3rd grade (CE2) . . . . . . . . . . . . . . . . . . 506.3 Evaluation at the entry into 6th grade . . . . . . . . . . . . . . . . . . . . . . 54

7 Conclusion: Discussion and limits 60

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1 Introduction

Gender di!erences are now much smaller than they once were. In particular with respectto educational attainment, women have caught up and now outperform men. There arenow more female than male college graduates (Goldin, Katz and Kuziemko 2006). Alreadyin school, female students obtain better scores than males in many subjects, especially inlanguages and verbal knowledge (Ellison and Swanson 2010, Niederle and Vesterlund 2010).In France, 56% of students enrolled in higher education in 2009 were female (DEPP 2011).The superior performance of girls can already be detected in primary school, where girls areless likely to be obliged to repeat a class (Caille and Rosenwald 2006).

Still, and even though less than before, wage inequality between men and women remains.Several countries have or are planning to put in place policies to increase the share of womenin well-paid and responsible positions (Hausman, Tyson and Zahidi, eds 2010). For exam-ple, Germany is currently strongly debating a gender quota for governing bodies of largeenterprises and France’s legislation mandates an equal share of men and women on politicalparties’ electoral lists (Hausman et al., eds 2010).

One of the reasons for the remaining wage inequality between men and women is theconsiderable underrepresentation of the latter in well-paid professions, in particular amongengineers. In 2008, only 27% of French engineering graduates were female (DEPP 2011). Theunderrepresentation of women is particularly consequential in this field given that engineersare strongly needed and thus are not only well paid, but also easily find a job.

However, the choice of most girls not to become engineers is already determined muchearlier than what one would think. While they outperform boys in most other subjects,girls lag behind in mathematics and other science-related subjects already towards the endof primary school. Indeed, there is a link between primary school mathematics achievement,the choice of a science-related career and gender wage inequality. As Niederle and Vesterlund(2010, p. 130) point out, mathematics scores in school are a good predictor for future earnings.Hence, a discussion of gender wage inequality must be based on a profound understanding ofthe evolution of the mathematics gender gap among young children.

While the causes of the gender gap in mathematics have been studied for more than30 years (see for example Fennema and Sherman 1977), its importance is nowadays notunanimously recognized. Declining over time, many current authors consider the gap to besmall (Penner 2008, Guiso, Monte, Sapienza and Zingales 2008). Hyde, who developed the“gender similarity hypothesis”, is convinced of the absence of a gender gap in mathematics(Hyde, Lindberg, Linn, Ellis and Williams 2008), however their sample does not follow thesame students over all ages. Recent research by Fryer and Levitt (2010) has brought themathematics gender gap back into attention, stating that girls score on average one fifth ofa standard deviation below boys at the end of primary school.

While these studies examine the gender gap at the mean, the existence of gender di!er-ences at the extremes of the distribution is more widely acknowledged. Benbow and Stanley(1980, 1983) suggest that the variance of girls’ mathematics scores is smaller than for boys.Therefore, boys are more represented at the upper and lower end of the distribution. Ellisonand Swanson (2010) examine the mathematics gender gap for the highest achieving studentsand find that girls are strongly underrepresented among the highest percentiles of students.This seems to be a specific feature of mathematics tests, because such a gap at the highend of the distribution does not exist in tests assessing general skills or reading (Ellison andSwanson 2010, p. 127).

When analyzing data from the Program for International Student Assessment (PISA),Guiso et al. (2008) find that the gender gap in mathematics test scores increases with quan-tiles. According to Niederle and Vesterlund (2010, p. 129), the fraction of female students inthe top five percent has not increased over the last 20 years and remains at one third. Ellisonand Swanson (2010, p. 110) make two observations concerning the gender gap at the top

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of the distribution. First, the gender gap considerably increases at the highest percentiles.Second, the origin of high-achieving girls and boys is not the same. While high-achievingboys come from various backgrounds, high-achieving girls are almost exclusively enrolled inthe most elitist schools of the US.

For a long time, it was not clear at which age the gender gap arises. Hyde, Fennema andLamon (1990) perform a meta-analysis of 254 e!ect sizes and find that the advantage of boysover girls in mathematics can first be observed in high-school. Leahey and Guo (2001) cometo the same conclusion. Fryer and Levitt (2010) are among the first to examine the gendergap in early years of school. Using the Early Childhood Longitudinal Study KindergartenCohort, they find that girls fall behind as early as 3rd grade of primary school. In 3rd grade,they score 0.2 standard deviations below boys, a similar number to the one we will detectin this study. A few decades ago, the weaker performance of girls in mathematics could beexplained with the fact that girls enrolled in less demanding mathematics classes than boys.Today, girls take equally demanding mathematics classes (Spelke 2005, p. 955). This is evenmore the case in primary school, as students generally do not have the choice about theclasses they follow. Given that girls score better than boys in most performance indicatorsother than mathematics, it is surprising that the gender gap in mathematics persists.

Most explanations on why girls score lower than boys in mathematics in spite of higherperformance everywhere else refer either to biological or sociological explanations. Penner(2008), as well as Fryer and Levitt (2010), give an overview of the most important theories.Researchers who explain the fact that boys score better than girls in mathematics withbiological or genetic di!erences justify their findings with di!erences in brain composition,hormone levels or inherent abilities. Proponents of environmental or sociological theoriesattribute the gender gap to particularities of the test setting as well as influences from parentsand teachers. One of their most important arguments points out that the education ofboys and girls is still strongly driven by stereotypes, which explain why girls score lower inmathematics.

The aim of this paper is to examine the evolution of the gender gap as well as the factorsthat contribute to its emergence in French primary school. The French Primary SchoolPanel 1997 (Panel de premier degre) from the French Ministry of Education provides gooddata to do so (DEPP 1997). It follows almost 10,000 students from schools all over France,who entered primary school in 1997 and collects detailed information about their background,family life and school career. Three standardized evaluations at the entry into primary schoolas well as at the entry into 3rd and 6th grade assess the evolution of mathematics and Frenchachievement for each student.

In addition to working with French data, this paper extends the recent work of Fryer andLevitt (2010) and Sohn (2010) by analyzing the determinants of the gender gap for sub-scoresof mathematics achievement such as geometry, problem solving and numeration. Examiningthe gender gap for sub-scores of mathematics achievement can be particularly beneficial giventhat the French Primary School Panel contains scores for skills such as technical knowledge,spatial skills, knowledge of temporal concepts. Among other things, this makes it possible toquestion the commonly stated theory according to which gender di!erences in spatial skillsare at the origin of the mathematics gender gap.

In their work on the racial gap, Fryer and Levitt (2004, 2006) find that black-white testscore di!erences can be strongly reduced by including a small number of explanatory vari-ables. Fryer and Levitt (2010) highlight that this is not the case for the gender gap, becausethe gender gap is not simply due to di!erences in background characteristics. Taking thisfinding as a starting point, I perform decompositions with the methods proposed by Oaxaca(1973) and Blinder (1973). Detailed decompositions allow to distinguish between the partof the gender gap which is due to gender di!erences in the endowment of each explanatoryvariable, gender di!erences in how each variable influences achievement as well as entirely

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unexplainable gender di!erences. While Fryer and Levitt (2010) only examine whether thetotal gender gap reduces once they control for background characteristics, I analyze how thegap changes due to each type of family, school and achievement input.

As stated before, many authors have found evidence that the gender gap varies within thedistribution and is more extreme at the higher end. Decomposition methods are also availablefor quantiles (Fortin, Lemieux and Firpo 2010). Sohn (2010) has published a study in whichhe performs quantile decompositions with the data used by Fryer and Levitt (2010). He findsthat the gender gap cannot be explained with di!erences in background characteristics, whichis not surprising given that background characteristics do not significantly di!er betweengenders. Unfortunately, the detailed decomposition cannot easily be realized with quantilesbecause it is based on an additive model structure. Given the choice between an aggregatequantile decomposition and a detailed decomposition at the mean, I choose the latter, becauseit allows to identify the distinct role of di!erent types of school and family input. Nonetheless,Section 5 is dedicated to describing the gender gap for the entire distribution at di!erent ages.

The detailed decomposition is based on a number of assumptions, which are often notsu"ciently discussed by researchers who use this method (Fortin et al. 2010). In particular,explaining the gender gap with a decomposition supposes the existence of an cognitive skillproduction function relating school and family input into educational achievement. Therefore,I develop a model based on the cognitive skill production function model by Todd and Wolpin(2003), which makes use of the rich data available in the French Primary School Panel. Giventhat all individual-specific and family background variables were collected only at one date,it is impossible to use panel data methods to eliminate the endogeneity bias resulting fromunobserved ability. However, the French Primary School Panel contains an evaluation of non-cognitive skills consisting of 15 items. As explained in Section 3, all regression estimates ofthis paper are therefore based on the identification assumption that the score of non-cognitiveskills evaluated for each student at the entry into primary school accounts for unobservedability. Section 3 develops and discusses the model underlying the analysis of this paperwith its assumptions and integrates it into the framework of the detailed Oaxaca-Blinderdecomposition.

The findings of this paper can be summarized in five main points. First, similar to Fryer andLevitt (2010) and consistent with findings from Caille and Rosenwald (2006, p. 127), I detectthe gender gap in French and mathematics. While the advantage of girls in French alreadyexists at the beginning of school and is stable through time, the gender gap in mathematicsarises between 3rd and 6th grade.

Second, the gender gap at the top of the distribution is smaller and develops earlier thanat among the lowest percentiles. This leads to a reversal of the distributional pattern ofthe gap. In 3rd grade, girls at the top of the distribution score approximately 0.05 to 0.1standard deviations lower than boys, whereas no gender di!erence exists among the lowestpercentiles. In 6th grade, the gender gap at the top of the distribution slightly increases to0.15 standard deviations, whereas girls score almost one third of a standard deviation belowboys at the lower end of the distribution. Hence, at the end of primary school, the gender gapdecreases with quantiles. This contradicts findings by authors such as Ellison and Swanson(2010), who claim that girls are particularly underrepresented among the best students.

Third, the decomposition shows that di!erences in observable characteristics such as familyand school inputs as well as prior achievement predict an advantage in mathematics for girls,which is particularly strong at the entry into school. In 1st, and to a lesser extent in 3rd grade,boys compensate their disadvantage in terms of endowments by having higher returns in termsof mathematics achievement to the same level of family inputs and non-cognitive skills. In 6thgrade, girls are predicted to score higher than boys both due to better endowments and higherreturns to family and school inputs, prior achievement as well as non-cognitive skills. Theactually observed advantage of boys is thus entirely unexplainable with the input variableswe have at our disposal.

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Fourth, I find no evidence in support of the theory according to which boys score betterin mathematics because they have better spatial skills (Niederle and Vesterlund 2010). Inthe evaluation of spatial skills at the entry into 1st grade, boys and girls obtain the samescores, the latter even outperform the former at the lowest percentiles. Moreover, my find-ings contradict the results of Leahey and Guo (2001), who state that the gender gap isstrongest in geometry. Leahey and Guo (2001, p. 714) claim that boys score particularlywell compared to girls in evaluations examining quantitative reasoning ability and analyticspatial-visualization ability. Citing numerous other studies, they point out that boys are par-ticularly well-performing in mathematical reasoning rather than mathematical computationand in geometry rather than algebra. This paper evaluates mathematics achievement for tensub-scores at the entry into 6th grade and finds that the gender gap is particularly low ingeometry. In 6th grade, girls score only 0.06 standard deviations below boys in geometry,whereas the average mathematics gender gap attains almost a quarter of a standard devi-ation. Moreover, in 3rd grade, girls even score higher than boys on average in geometry.Cross-country di!erences may be a possible explanation. Penner (2008, p. S162) points outthat distributional gender di!erences vary from country to country.

Finally, this paper provides some support for theories attributing the mathematics gendergap to stereotypical thinking. Working with the same data, Caille and Rosenwald (2006,p. 127) explain the French gender gap in mathematics and France with stereotypes and justifytheir theory with evidence stating that girls underestimate the mathematics skills comparedto their actual achievement. With regards to stereotypical thinking of parents, I find thatthe opinion of parents about their child’s achievement in French and mathematics shows thetypical gender pattern even before gender di!erences arise in actual scores. Moreover, theopinion of parents about their child’s mathematics achievement is strongly correlated withactual scores. A clear proof of stereotypes as a reason for the gender gap in mathematics is,however, not possible, as I have no way to verify the direction of the causality.

Before immersing ourselves into the empirical analysis, I thoroughly develop the theoreticalframework underlying the examination of the mathematics gender gap. Section 2 summarizesthe biological and sociological theories which have been provided as an explanation of genderdi!erences in mathematics achievement. Section 3 develops the econometric strategy, whichwill allow us to decompose the gender gap into its explainable and structural components.With the theoretical framework in mind, Sections 4 to 6 compose the empirical part of thispaper. Section 4 presents the French Primary School Panel. Section 5 describes the gendergap in mathematics and French for the entire distribution, as it develops throughout 1st,3rd and 6th grade. Finally Section 6 presents the results of the detailed Oaxaca-Blinderdecomposition.

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2 Genes or stereotypes? – Possible explanations for thegender gap in mathematics

Table 1: Overview of the theories explaining the gender gap in mathematics (based on Fryerand Levitt 2010)

Field Theory

Biological Di!erences in brain composition (Cahill 2005)Di!erences in hormone levels (Davison and Susman 2001)Di!erences in strategy (Fennema and Carpenter 1998)Di!erences in spatial ability (Niederle and Vesterlund 2010)

Sociological Boys are better in timed settingsWording of questions plays a roleBoys are better in competitive settings (Niederle and Vesterlund 2010)Math anxiety (Ho et al. 2000)Di!erential parental treatment and expectations(Fryer and Levitt 2010)Di!erential treatment by teachers (Niederle and Vesterlund 2010)Stereotypes (Duru-Bellat 2010)

Many felt provoked when Larry Summers (2005), in his remarks at the NBER “Conferenceon diversifying the Science and Engineering workforce”, claimed that a gender test score gapexisted due to genetic di!erences. He justified his argument by stating that girls systemat-ically score with a lower variance than boys. Therefore, girls are underrepresented amongthe excellent. Despite the indignation following Summer’s remark, biological or genetic argu-ments were cited among the first attempts to explain the gender gap in mathematics. Authorsin these fields highlight di!erences between boys and girls with respect to brain composition,hormone levels, strategy and spatial ability (Fryer and Levitt 2010).

Some proponents of biological theories of the gender gap provide explanations which iden-tify a clear causal link between the construction of male and female brains and their respectiveachievement. Cahill (2005) presents evidence from experiments showing how boys and girlsprocess the same information di!erently in their brain. Therefore, he claims, sociologicaltheories alone cannot explain gender di!erences. In an experimental study, Kucian, Loen-neker, Dietrich, Martin and Aster (2005) examine the activity of di!erent brain regions whilesolving arithmetic and geometric problems. They find that men and women use di!erentparts of their brain and thus apply di!erent strategies when solving these problems. Froma more descriptive point of view, Fennema and Carpenter (1998) find that primary schoolboys and girls pursue di!ering strategies when solving simple numerical problems. Whileboys use more abstract solution methods based on understanding the material, girls solve thequestions with concrete calculations.

While these explanations claim a direct link between brain composition and mathematicsability, other biological theories are more complementary with sociological arguments. Davi-son and Susman (2001) relate sex di!erences in cognitive ability to di!ering hormone levels.With children aged 9 to 15, they test and validate the hypothesis according to which higherlevels of testosterone are associated with better spatial skills. As the authors point out them-selves, their evidence can be considered as complementary with sociological explanations,because the relation between hormones and cognitive ability may not be causal. Rather,children with particular hormone levels might be more likely to experience di!erent ways ofchild development or be exposed to typical activities.

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Probably the oldest and most often cited theory on the gender gap in mathematics statesthat boys are equipped with superior spatial skills than girls. As an example of recentliterature, Lawton and Hatcher (2005) conduct experiments evaluating the visuospatial short-term memory of men and women. Men were able to memorize shapes and their positionmore accurately. Niederle and Vesterlund (2010, p. 130) point out that gender di!erencesin spatial ability already develop at young age. A possible explanation states that thesedi!erences evolve due to di!erent ways of playing. According to this theory, boys are moreoriented towards activities that involve moving. Moreover, already at very young age, boyslearn about objects while girls learn about persons and emotions (Spelke 2005).

Spelke (2005) criticizes theories explaining the gender gap with biological or genetic dif-ferences between boys and girls. She claims that these findings are not valid because theyare based on experiments, which could also have been interpreted di!erently. Indeed, exper-iments such as the one conducted by Kucian et al. (2005), are often realized with a smallnumber of participants. Moreover, Spelke (2005) cites studies according to which very youngboys and girls perform equally well in numerical and geometric tasks. Hence, children do notpossess any predetermination for mathematics ability. A similar argument is put forward byHyde (2005) with the development of her “gender similarity hypothesis”.

Other researchers explain the gender gap in mathematics with di!erences that can be at-tributed to sociological and environmental influences received by children. Theories justifyinggender di!erences with the way tests are administered examine the reluctance of girls towardscompetitive settings, the wording of test questions, di!erential performance in timed testsor simply math anxiety (Fryer and Levitt 2010). Theories identifying gender di!erences inmathematics ability itself analyze how the child grows up in society, how she or he is influ-enced by parental treatment and expectations. Moreover, gender di!erences may arise dueto di!erential treatment by teachers and stereotypical threats.

Some sociological theories explain gender di!erences in mathematics performance with themodalities of the examination itself. They state that the wording of test question disad-vantages girls. Moreover, girls might score lower because they are less comfortable withtimed settings (Fryer and Levitt 2010). Ho, Senturk, Lam, Zimmer, Hong, Okamoto, Chiu,Nakazawa and Wang (2000) claim that girls su!er more strongly from mathematics anxietythan boys. Niederle and Vesterlund (2010) question the conclusion that lower test scores ofgirls are an indicator of weaker mathematics ability by explaining that girls are reluctant toenter into a competitive setting. This might either be due to the fact that they have a lowerself-confidence when competing against male students or that girls are more averse towardsrisk. Many authors have analyzed these two phenomena.1 Therefore, observed gender dif-ferences in mathematics test scores cannot be attributed to di!erent capabilities of boys andgirls, but rather to the fact that competitive tests do not reflect actual di!erences in abilities.They remind that test score results not only depend on cognitive, but also on non-cognitivefactors (Niederle and Vesterlund 2010, p. 136).

In addition to theories attributing gender di!erences to the test situation, there are con-vincing sociological arguments highlighting actual di!erences in mathematics ability. Theyexplain the mathematics gender gap with environmental factors influencing the developmentand mathematics ability of children. These influencing factors can be parental expectations,teacher influence and general stereotypes in the society, which are transferred to childrenthrough parents, teachers and other people. Fryer and Levitt (2010) consider the possibilitythat mothers employed in mathematics-related occupations might have a particular expec-tation towards their child’s mathematics achievement. However, they find no evidence tosupport this theory. Moreover, Niederle and Vesterlund (2010, p. 137) refer to studies show-ing that the teacher’s gender influences the performance of female students. A gender gap inmathematics achievement could therefore be explained by the fact that the share of femaleteachers is much lower in mathematics than in any other subject.

1For an extensive literature review, see Niederle and Vesterlund (2010)

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An often cited theory relates the gender gap in educational achievement to the still dom-inating stereotype according to which mathematics is for boys and language is for girls.Stereotypical patterns are very strong at all age levels. In their examination regarding thechoice of university major and labor market situation as well as the choices made for privatelife, Benbow, Lubinski, Shea and Eftekhari-Sanjani (2000) find stereotypical gender patternseven among adults who were among the highest percentile of mathematics achievers whenthey were 11 to 13 years of age.

Pope and Syndor (2010) find significant di!erences in the magnitude of the gender gapbetween US states and census divisions. They examine a strong correlation between thegender gap in math and in reading. In states, in which boys strongly outperform girls inmath, girls strongly outperform boys in reading. Thus, a stronger gap cannot be interpretedas a the existence of a more boy- or girl-friendly learning environment, but rather showsthat the gender gap is related to stereotypes (Pope and Syndor 2010, p. 101). Based on themagnitude of the gender gap, Pope and Syndor construct a stereotype index for each state.They find that this stereotype index is negatively correlated with median income. They alsodetect a strong correlation between the stereotype index and the probability of people sharingthe opinion that women should take care of homework while men should earn money outsidethe home in the General Social Survey. The same is true with the answers to the questionwhether “mathematics is for boys” (Pope and Syndor 2010, p. 106).

Gender di!erences about stereotypes are present even in the classroom itself. Duru-Bellat(2010) attributes an important part of the gender di!erence to the fact that boys and girlsstudy in mixed classes. She points out that girls score particularly low in mathematics in thepresence of boys. Separating genders would allow girls to have a higher confidence in theirabilities and to be less subject to the stereotype according to which they are supposed toscore lower than their male classmates.

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3 The econometric model

This paper aims to examine the influence of di!erent family and school inputs on the emer-gence of a gender gap in mathematics in primary school. In order to assess their impacton the gender gap, it is important to establish a theory linking these inputs to evaluationsof mathematics achievement. In this context, it is natural to think within the frameworkof educational achievement production function model, as formalized by Todd and Wolpin(2003), which is widely used in economics of education.

Todd and Wolpin (2003) formalize and combine two approaches which are explicitly orimplicitly underlying all quantitative research on educational achievement. On the one hand,the early childhood development approach studies the role of parental background character-istics and home environment in determining educational achievement. One the other hand,proponents of the education production function theory seek to explain the development ofcognitive skills with school inputs such as school quality and the influence of teachers. Relat-ing inputs to a measurable output – the test score – both theories draw the parallel betweenthe acquisition of knowledge and the production process in a firm.

This paper aims to take Todd and Wolpin’s (2003) model as a starting point. I develop asimilar model which may apply to the data from the French Primary School Panel. Using thismodified version of the Todd and Wolpin (2003) model, it is possible to carry out a detailedOaxaca-Blinder decomposition of the gender gap in mathematics, which arises between 3rdand 6th grade. The decomposition aims to distinguish three components of the gender gap.First, I will predict the magnitude of the gender gap, which we would observe, if genderdi!erences were exclusively due to di!erences in school and family inputs. Second, this paperwill be able to identify the part of the gender gap, which can be explained by gender di!erencesin transforming inputs into achievement. Finally, having distinguished these components, theremaining gender gap can be considered as purely unexplainable.

The Todd and Wolpin (2003) model, as well as the detailed Oaxaca-Blinder decomposition,rely on a number of identifying assumptions. On the one hand, assumptions are necessaryin order for the model itself to be valid. One the other hand, additional assumptions arerequired for the estimation of the decomposition itself. Being clear about the underlyingassumptions is necessary in order to be certain of the parameters estimated. The remainderof this section therefore focuses on the description of the model and the decomposition aswell as the assumptions required for them to be valid. It will end with a presentation of theempirically estimated equations.

3.1 A economic model derived from Todd and Wolpin

Formulating human capital accumulation as a production function process goes back to Ben-Porath (1967). Complex models are now used to explain the acquisition of cognitive andnon-cognitive skills as a function of di!erent input types. According to Cunha and Heckman(2008), who formalized the development of non-cognitive skills, the state of the art approachof modeling cognitive skill formation is the model proposed by Todd and Wolpin (2003).Todd and Wolpin (2003) present a simple two-period dynamic model aiming to reunite twobranches of the literature which seek the determinants of educational outcomes either infamily or in school input.

Figure 1 summarizes Todd and Wolpin’s (2003) production function model of cognitiveskill formation. The figure distinguishes between three components, which can be measuredin di!erent time periods. Educational achievement measures are situated on the left of thefigure. Family input is shown in the middle and school input is represented on the right.Arrows indicate the interdependencies between achievement as well as family and schoolinput. For better readability, they look di!erently depending on where they originate. Anarrow indicates that its component of origin influences the component to which it points.

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t0

Achievement in t1t1

t2

A1 = g0(F0, µ) (5)

School achievement

Achievement in t2

A2 = g1(F0, S1, F1, µ) (6)

Family input School input

Family input in t0

F0 = !0(X,µ) (1)

school input in t1 as influenced by parents

actual school input in t1

Family input in t1

F1 = !1(A1, X, µ, S1 ! S1) (2)

S1 = !1(A1, µ) (3)S1 = !1(A1, X, µ) (4)

Figure 1: Production function modeling according to Todd and Wolpin (2003)

The first period, denoted t0, is the period prior to the entry into school. Here, F0 repre-sents family background characteristics as well as family inputs before the entry into school.Unobservable family characteristics are approximated with observable values X, such as theeducation and profession of parents. Moreover, µ represents the innate ability of the child,which is supposed to be predetermined from birth and invariant in time. Presenting ini-tial family input F0 as a function of both X and µ allows for the possibility that the inputprovided by parents may depend on the innate ability of the child.

Todd and Wolpin (2003) present the first period achievement measure, denoted A1, as afunction of family input prior to school, F0, as well as the innate ability of the child, µ. AsA1 is supposed to be determined at the beginning of period 1, it has an impact on familyand school input in this period rather than being influenced by them. Todd and Wolpin(2003) distinguish between actual school input and school input as influenced by the parents.The former, S1, contains elements which apply to all students of a school, such as measuresof school quality and teacher input. As can be seen in Figure 1, these inputs are supposedto be a function of achievement at the beginning of the period as well as innate ability. Inmy view, it is not entirely clear why school input should depend on these factors, but Toddand Wolpin (2003, p. F8) justify their model by stating that the school might “choose[] inputlevels for a particular child purposefully, taking into account the child’s achievement level andthe endowment”. School input as influenced by parents, denoted as S1, is justified mainly aslocation decisions or whether the child attends a private or a public school. Together withactual school input and achievement at the beginning of the period, it influences family inputin period 1, denoted by F1. Note that both family input, F1, and school input as influencedby parents, S1, also depend on family background characteristics, X.Achievement in period 2, denoted A2, is a function of all former inputs. As shown in

Figure 1, it is a function of family input prior to school, F0, actual school and family input inperiod 1, S1 and F1 as well as the students innate ability. Note that school input as influencedby parents modifies family input, but does not have a direct impact on achievement in thefollowing period, whereas actual school input does. Moreover, according to Todd andWolpin’s(2003) model, achievement in period 1 a!ects achievement in the following period only viaits potential influence on school and family inputs and not directly. One might criticize thisconstruction of the model, given that both achievement measures could be directly related

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through the acquisition of non-cognitive skills. If a particular level of achievement in period 1contributes to the student’s self-confidence, which improves his achievement in period 2, therewould be a direct influence which does not pass via school or family input. Moreover, suchan impact is not captured in the term µ, because µ is assumed to be time invariant. Hence,Todd and Wolpin’s (2003) model potentially underestimates the role of non-cognitive skills.Based on models in the spirit of Todd and Wolpin (2003) and their predecessors, proponents

of the Early Childhood Development as well as the Education Production Function literatureheavily debate which factors influence educational achievement. They find di!ering resultseven by using the same data (Todd and Wolpin 2003, p. F5). Todd and Wolpin (2003) statethat such disagreements are due to di!ering econometric specifications. Indeed, estimatingthe model as depicted in Figure 1 is complicated by the fact that individual ability is un-observable and that data will often be limited. Therefore, Todd and Wolpin (2003) proposedi!erent specifications depending on data availability. For each of them, they thoroughlystate the necessary assumptions in order for the strategy to be valid.The contemporaneous specification allows to estimate the model in case the researcher

has to rely on input from one time period only. In this case, only current explanatory variables(school and family input) are included to estimate current achievement. In order for thecontemporaneous specification to be valid, the following assumptions are made:

1. Only contemporaneous inputs matter for explaining current achievement or

2. Inputs do not change over time, hence contemporaneous inputs capture the entire his-tory of inputs

and

3. Contemporaneous inputs are unrelated to unobserved characteristics.

It is very unrealistic to assume that only contemporaneous inputs matter for educationalachievement or that they capture the entire history of inputs (Todd and Wolpin 2003, p. F16).Moreover, the contemporaneous specification requires inputs to be unrelated to unobservedcharacteristics such as the innate ability represented as µ in Figure 1 (assumption 3). Ifwe believe in the model, the third assumption is by definition wrong, because we assumedinnate ability µ to influence school and family input. A more reliable estimation strategythus cannot be implemented without the inclusion of historical input measures.The value-added specification is similar to the contemporaneous specification, but ad-

ditionally includes a baseline achievement measure. By including a baseline achievementmeasure, the value-added specification is supposed to capture the child’s unobserved ability.Todd and Wolpin (2003) highlight the following assumptions, which have to be made in orderfor the value-added specification to be valid:

1. The educational achievement production function must be non-varying with age.

2. The coe"cients relating input to educational achievement must be geometrically de-clining with distance.

3. The impact of innate ability on educational achievement must be geometrically decliningat the same rate as input e!ects.

4. The baseline achievement measure accounts for all unobserved ability.

The main role of these assumptions is to allow the value-added specification to be used in adynamic setting, i.e. to use it in order to estimate achievement for many time periods. In thiscontext, the first assumption ensures that the lagged achievement measure is permanentlyable to capture unobserved abilities. For the purpose of this paper, this restriction is notimportant because educational achievement is estimated at three precise moments in time,which means that variations in the exact educational achievement production function areunproblematic. The second and third assumption also aim to validate the model for a dynamic

12

setting including many time periods. Moreover, they provide a justification for not having toinclude lagged school and family input in the estimation, as their impact is already capturedby the lagged achievement measure.

Assumption 3 is a somewhat artificial assumption. Todd and Wolpin (2003) highlight someimplications of the value-added specification for unobserved characteristics if this assumptionis not true. If we allow unobserved characteristics to vary over time, unobserved character-istics at a later age are likely to be correlated with prior input as well as prior achievement.However, if this is the case, the zero conditional mean assumption – necessary in order toobtain a valid detailed decomposition – is no longer verified. Thus, if we apply the value-added specification, we must assume that unobserved characteristics are a fixed input, whichis determined prior to the entry into primary school and does not change over time.

Furthermore, as stated in assumption 4 and by the definition of the model, the value-added achievement measure depends on unobserved ability. Thus, even if unobserved abilityis assumed to be constant over time, the baseline achievement measure will be endogenous.Todd and Wolpin (2003) propose to eliminate the unobserved factor in the determination ofthe baseline achievement measure by taking the di!erence between two baseline achievementmeasures from two previous time periods. The drawback of such a di!erenced achievementmeasure is, of course, that a further period of data is required, which brings us closer to themost complete estimation proposed by Todd and Wolpin (2003).

The cumulative specification explains educational achievement with all current and pastfamily and school inputs. As this specification is very close to the actual model presentedabove, it is based on only one, but crucial assumption: Included inputs (contemporaneous andhistorical family and school inputs) are uncorrelated with omitted inputs and measurementerror in test scores. If one believes that this assumption does not hold, Todd and Wolpin(2003) propose to eliminate unobservable ability by using family or siblings fixed e!ects.

The choice of applying one of Todd and Wolpin’s (2003) model estimation methods stronglydepends on the data that is available to the researcher. Given the data from the FrenchPrimary School Panel available for this paper, I consider it useful to specify a model whichis a realistic approximation for the achievement production process of the students in oursample. Figure 2 presents the model which will serve as the theoretical foundation of theanalysis in this paper. It looks complicated at first, but is actually very close to the model byTodd and Wolpin (2003). Given that this paper aims to examine data from three standardizedevaluations (at the entry into 1st, 3rd and 6th grade), the model is set up for these threetime periods in addition to one time period before the entry into school. Again, input typesdistinguish between family and school and arrows indicate the dependencies between inputsand achievement.

In addition to Todd and Wolpin’s (2003) model, family input before the entry into primaryschool, in period 0, is complemented by maternal school input. Family input before the entryinto primary school comprises family background characteristics such as family composition,living conditions as well as the age, nationality, education and profession of both parents.Some of these variables were collected from the parent questionnaire, which was administeredin the second year of primary school. However, it is a relatively uncritical assumption toconsider these background characteristics as time invariant and therefore state that they arevalid as proxies for family input before the entry into primary school. As maternal schoolinput, we consider the duration of maternal school attendance as well as the opinion of thechild’s parents concerning their child’s gains and pleasure of attending maternal school. Bothfamily and maternal school input is denoted as F0.

As in Todd and Wolpin’s (2003) model, achievement at the entry into primary school,denoted A1 is influenced by family and maternal school input F0. The second di!erence totheir model is that I assume achievement A1 to additionally depend on a measure of non-cognitive skills, denoted NC, which is supposed to capture the influence of the child’s ability,

13

t0

Evaluation at entry into primary school

Determined by no factors other than prior family

and elementary school input

Variables: General, Maths, French

t1

t2

School achievement Family input School input

Family and elementary school input before primary school (denoted )

Initial conditions, they are not influenced by subsequent school outomces

Variables: with whom child lives, family composition, information on parents (age, nationality, language, education, job), living conditions, childcare, elementary school information, parents‘ opinion about elementary school

The empirical model

Elementary school input

E0 (1)

Family input (Early Childhood Development)

F0 (2)

F1 = !1(A1, X, µ, S1 ! S1) (3)

School input (Education Production Function)

S1 = "1(A1, µ) (4)

S1 = #1(A1, X, µ) (5)

School achievement

A1 = g0(F0, µ) (6)

A2 = g1(F0, S1, F1, µ) (7)

Variables

Ft – Family input year t, determined from production function !t

St – School input in year t, determined from production function "t, no school input in year 0, becausefamily inputs predate school inputs

St – School input as influenced by parents, determined from production function #t

At – achievement in year t, determined from production function gt!1, because it depends on theinput factors determined in t! 1

X – observable family background characteristics

µ – unobservable individual characteristics of child (e.g. ability)

Assumptions

1. Production function approach

2. Family and school inputs have an impact on achievement in the following year

1

Family input after entry into primary school

1. Exogenously determined, comparable toVariables: Distance to school, who is in charge of child

2. Determined by onlyVariables: public/private school, child‘s activities

3. Determined by and Variables: contact with school, parents‘ opinion on teacher/child, help with homework

The empirical model


E0 (1)


F0 (2)

F1 = !1(A1, X, µ, S1 ! S1) (3)


S1 = "1(A1, µ) (4)

S1 = #1(A1, X, µ) (5)

School achievement

A1 = g0(F0, µ) (6)

A2 = g1(F0, S1, F1, µ) (7)

Variables







Assumptions



1

The empirical model


E0 (1)


F0 (2)

F1 = !1(A1, X, µ, S1 ! S1) (3)


S1 = "1(A1, µ) (4)

S1 = #1(A1, X, µ) (5)

School achievement

A1 = g0(F0, µ) (6)

A2 = g1(F0, S1, F1, µ) (7)

Variables







Assumptions



1

The empirical model


E0 (1)


F0 (2)

F1 = !1(A1, X, µ, S1 ! S1) (3)


S1 = "1(A1, µ) (4)

S1 = #1(A1, X, µ) (5)

School achievement

A1 = g0(F0, µ) (6)

A2 = g1(F0, S1, F1, µ) (7)

Variables







Assumptions



1

The empirical model


F0 (1)

F1 = !1(A1, X, µ, S1 ! S1) (2)


S1 = "1(A1, µ) (3)

S1 = #1(A1, X, µ) (4)

School achievement

A1=a0 + F !0a1! "# $+ $1!"#$

A1 $1(5)

A2 = g1(F0, S1, F1, µ) (6)

Variables




At – achievement in year t, determined from production function gt"1, because it depends on theinput factors determined in t! 1



Assumptions



1

The empirical model


F0 (1)

Fj1 (2)

Fk1=m0 + F !0m1! "# $+ !k1!"#$

Fk1 !k1(3)

Fl1=n0 + F !0n1 + [A!

1n2 + "!1n3]! "# $+ !l1!"#$Fl1 !l1

(4)


S1 = #1(A1, µ) (5)

S1 = $1(A1, X, µ) (6)

School achievement

A1=a0 + F !0a1! "# $+ "1!"#$

A1 "1(7)

A2 = g1(F0, S1, F1, µ) (8)

Variables

Ft – Family input year t, determined from production function %t

St – School input in year t, determined from production function #t, no school input in year 0, becausefamily inputs predate school inputs

St – School input as influenced by parents, determined from production function $t




Assumptions



1

School input in CP & CE1

Determined exogenously (even

though correlated with family inputs)

Variables: nb of students total and by

foreign/repeaters/gender, rural/urban

The empirical model


F0 (1)

Fj1 (2)

Fk1=m0 + F !0m1! "# $+ !k1!"#$

Fk1 !k1(3)

Fl1=n0 + F !0n1 + [A!

1n2 + "!1n3]! "# $+ !l1!"#$Fl1 !l1

(4)


S1 (5)

S1 = #1(A1, X, µ) (6)

School achievement

A1=a0 + F !0a1! "# $+ "1!"#$

A1 "1(7)

A2 = g1(F0, S1, F1, µ) (8)

Variables

Ft – Family input year t, determined from production function $t

St – School input in year t, determined from production function %t, no school input in year 0, becausefamily inputs predate school inputs





Assumptions



1

Evaluation at entry into CE2

Determined by all prior inputs

Variables: Mathematics (4 topics)

Evaluation at entry into sixième

Determined by all prior inputs, except and

Variables: Mathematics (5 topics, 5 skills)t3

School input in CE2, CM1, CM2

Determined exogenously (even though

correlated with family inputs)

Variables: same as above

The empirical model


F0 (1)

Fj1 (2)

Fk1=m0 + F !0m1! "# $+ !k1!"#$

Fk1 !k1(3)

Fl1=n0 + F !0n1 + [A!

1n2 + "!1n3]! "# $+ !l1!"#$Fl1 !l1

(4)

Fl1=n0 + F !0n1 +A!

1n2! "# $+ !l1!"#$Fl1 !l1

(5)


S1 (6)

S2 (7)

School achievement

A1=a0 + F !0a1! "# $+ "1!"#$

A1 "1(8)

A2=b0 + F !0b1 + F !

j1b2 + [F !k1b3 + !k1b4] + [F !

l1b5 + !l1b6] + [A!1b7 + "!1b8] + S!

1b9! "# $+ "2!"#$

A2 "2(9)

A2=b0 + F !0b1 + F !

.1b2 +A!1b3 + S!

1b4! "# $+ "2!"#$A2 "2

(10)

Variables

Ft – Family input year t, determined from production function #t

St – School input in year t, determined from production function $t, no school input in year 0, becausefamily inputs predate school inputs

St – School input as influenced by parents, determined from production function %t




1

The empirical model


F0 (1)

Fj1 (2)

Fk1=m0 + F !0m1! "# $+ !k1!"#$

Fk1 !k1(3)

Fl1=n0 + F !0n1 + [A!

1n2 + "!1n3]! "# $+ !l1!"#$Fl1 !l1

(4)

Fl1=n0 + F !0n1 +A!

1n2! "# $+ !l1!"#$Fl1 !l1

(5)

Fk2 = r0 + F !0r1 + F !

j1r2 + !k2 (6)


S1 (7)

S2 (8)

School achievement

A1=a0 + F !0a1! "# $+ "1!"#$

A1 "1(9)

A2=b0 + F !0b1 + F !

j1b2 + [F !k1b3 + !k1b4] + [F !

l1b5 + !l1b6] + [A!1b7 + "!1b8] + S!

1b9! "# $+ "2!"#$

A2 "2(10)

A2=b0 + F !0b1 + F !

.1b2 +A!1b3 + S!

1b4! "# $+ "2!"#$A2 "2

(11)

A2 = b0 + F !0b1 + F !

.1b2 +A!1b3 + S!

1b4 + "2 (12)

Variables







1

The empirical model


F0 (1)

Fj1 (2)

Fk1=m0 + F !0m1! "# $+ !k1!"#$

Fk1 !k1(3)

Fl1=n0 + F !0n1 + [A!

1n2 + "!1n3]! "# $+ !l1!"#$Fl1 !l1

(4)


S1 (5)

S1 = #1(A1, X, µ) (6)

School achievement

A1=a0 + F !0a1! "# $+ "1!"#$

A1 "1(7)

A2 = g1(F0, S1, F1, µ) (8)

Variables

Ft – Family input year t, determined from production function $t

St – School input in year t, determined from production function %t, no school input in year 0, becausefamily inputs predate school inputs





Assumptions



1

The empirical model


F0 (1)

Fj1 (2)

Fk1 = m0 + F !0m1 + !k1 (3)

Fl1 = n0 + F !0n1 + [A!

1n2 + "!1n3] + !l1 (4)

Fl1 = n0 + F !0n1 +A!

1n2 + !l1 (5)

Fk2 = r0 + F !0r1 + F !

j1r2 + !k2 (6)


S1 (7)

S2 (8)

School achievement

A1 = a0 + F !0a1 +NC !a2 + "1 (9)

A2 = b0 + F !0b1 + F !

j1b2 + [F !k1b3 + !k1b4] + [F !

l1b5 + !l1b6] + [A!1b7 + "!1b8] + S!

1b9 + "2 (10)

A2 = b0 + F !0b1 +NC !b2 + F !

.1b3 +A!1b4 + S!

1b5 + "2 (11)

A3 = c0 + F !0c1 +NC !c2 + F !

.1c3 + F !k2c4 +A!

1c5 + S!1c6 + "3 (12)

Variables







1

The empirical model


F0 (1)

Fj1 (2)

Fk1 = m0 + F !0m1 + !k1 (3)

Fl1 = n0 + F !0n1 + [A!

1n2 + "!1n3] + !l1 (4)

Fl1 = n0 + F !0n1 +A!

1n2 + !l1 (5)

Fk2 = r0 + F !0r1 + F !

j1r2 + !k2 (6)


S1 (7)

S2 (8)

School achievement

A1 = a0 + F !0a1 +NC !a2 + "1 (9)

A2 = b0 + F !0b1 + F !

j1b2 + [F !k1b3 + !k1b4] + [F !

l1b5 + !l1b6] + [A!1b7 + "!1b8] + S!

1b9 + "2 (10)

A2 = b0 + F !0b1 +NC !b2 + F !

.1b3 +A!1b4 + S!

1b5 + "2 (11)

A3 = c0 + F !0c1 +NC !c2 + F !

.1c3 + F !k2c4 +A!

1c5 + S!1c6 + "3 (12)

Variables







1

The empirical model


F0 (1)

Fj1 (2)

Fk1 = m0 + F !0m1 + !k1 (3)

Fl1 = n0 + F !0n1 + [A!

1n2 + "!1n3] + !l1 (4)

Fl1 = n0 + F !0n1 +A!

1n2 + !l1 (5)

Fk2 = r0 + F !0r1 + F !

j1r2 + !k2 (6)


S1 (7)

S2 (8)

School achievement

A1 = a0 + F !0a1 +NC !a2 + "1 (9)

A2 = b0 + F !0b1 + F !

j1b2 + [F !k1b3 + !k1b4] + [F !

l1b5 + !l1b6] + [A!1b7 + "!1b8] + S!

1b9 + "2 (10)

A2 = b0 + F !0b1 +NC !b2 + F !

.1b3 +A!1b4 + S!

1b5 + "2 (11)

A3 = c0 + F !0c1 +NC !c2 + F !

.1c3 + F !k2c4 +A!

1c5 + S!1c6 + "3 (12)

Variables







1

The empirical model


F0 (1)

Fj1 (2)

Fk1 = m0 + F !0m1 + !k1 (3)

Fl1 = n0 + F !0n1 + [A!

1n2 + "!1n3] + !l1 (4)

Fl1 = n0 + F !0n1 +A!

1n2 + !l1 (5)

Fk2 = r0 + F !0r1 + F !

j1r2 + !k2 (6)


S1 (7)

S2 (8)

School achievement

A1 = a0 + F !0a1 +NC !a2 + "1 (9)

A2 = b0 + F !0b1 + F !

j1b2 + [F !k1b3 + !k1b4] + [F !

l1b5 + !l1b6] + [A!1b7 + "!1b8] + S!

1b9 + "2 (10)

A2 = b0 + F !0b1 +NC !b2 + F !

.1b3 +A!1b4 + S!

1b5 + "2 (11)

A3 = c0 + F !0c1 +NC !c2 + F !

.1c3 + F !k2c4 +A!

1c5 + S!1c6 + "3 (12)

Variables







1

The empirical model


F0 (1)

Fj1 (2)

Fk1 = m0 + F !0m1 + !k1 (3)

Fl1 = n0 + F !0n1 + [A!

1n2 + "!1n3] + !l1 (4)

Fl1 = n0 + F !0n1 +A!

1n2 + !l1 (5)

Fk2 = r0 + F !0r1 + F !

j1r2 + !k2 (6)


S1 (7)

S2 (8)

School achievement

A1 = a0 + F !0a1 +NC !a2 + "1 (9)

A2 = b0 + F !0b1 + F !

j1b2 + [F !k1b3 + !k1b4] + [F !

l1b5 + !l1b6] + [A!1b7 + "!1b8] + S!

1b9 + "2 (10)

A2 = b0 + F !0b1 +NC !b2 + F !

.1b3 +A!1b4 + S!

1b5 + "2 (11)

A3 = c0 + F !0c1 +NC !c2 + F !

.1c3 + F !k2c4 +A!

1c5 + S!1c6 + "3 (12)

Variables







1

The empirical model


F0 (1)

Fj1 (2)

Fk1 = m0 + F !0m1 + !k1 (3)

Fl1 = n0 + F !0n1 + [A!

1n2 + "!1n3] + !l1 (4)

Fl1 = n0 + F !0n1 +A!

1n2 + !l1 (5)

Fk2 = r0 + F !0r1 + F !

j1r2 + !k2 (6)


S1 (7)

S2 (8)

School achievement

A1 = a0 + F !0a1 +NC !a2 + "1 (9)

A2 = b0 + F !0b1 + F !

j1b2 + [F !k1b3 + !k1b4] + [F !

l1b5 + !l1b6] + [A!1b7 + "!1b8] + S!

1b9 + "2 (10)

A2 = b0 + F !0b1 +NC !b2 + F !

.1b3 +A!1b4 + S!

1b5 + "2 (11)

A3 = c0 + F !0c1 +NC !c2 + F !

.1c3 +A!1c4 + S!

2c5 +A!2c6 + "3 (12)

Variables




At – achievement in year t, determined from production function gt"1, because it depends on theinput factors determined in t ! 1



1

Figure 2: Modified version of the Todd and Wolpin (2003) model

denoted µ in Todd and Wolpin (2003). The child’s non-cognitive skills will be approximatedwith the teacher’s evaluation of non-cognitive skills provided in the French Primary SchoolPanel. In addition to capturing the child’s non-cognitive skills and unobserved cognitiveability, this assessment might be driven by stereotypical thinking of teachers vis-a-vis theabilities of girls and boys. We need to keep the limited accuracy of this variable in mind.The third di!erence with respect to Todd and Wolpin (2003) is that in the modified model,

achievement at the entry into primary school does not influence school input, denoted S1.This assumption simply takes into account that variables on school input from the FrenchPrimary School Panel are not student-specific. Family input after the entry into primaryschool comprises variables which may potentially change over time and were obtained fromthe parent questionnaire in 2nd grade. In Figure 2, family input after the entry into schoolis grouped into three types according to the dependency on formerly determined variables.Variables such as the distance between home and school and who is in charge of the child areconsidered to be independent of former input and achievement and are denoted as Fj1. Othervariables, such as the child’s free-time activities and the choice between public and privateschool are considered to depend on previous family input, F0, but not on prior achievementand are denoted as Fk1. Finally, parents’ contact with primary school, their opinion aboutthe teacher and their child’s school achievement as well as their support with the child’shomework is assumed to depend both on former family input F0 and achievement at the entryinto school, A1. The di!erence between these types of family input is not very important forthe estimation, but should be kept in mind for the interpretation of the results.

Achievement in time period 2, corresponding to the standardized evaluation at the entryinto CE2 (3rd grade), is assumed to depend on all inputs and achievement determined up tothat date. Like in the original model, it is denoted A2. The fourth main di!erence to Todd

14

and Wolpin (2003) is that I assume achievement in 3rd grade to be influenced by achievementat the entry into primary school, A1. This assumption aims to translate the fact that formerachievement might contribute to the development of non-cognitive skills and thereby influenceachievement in subsequent cognitive skill evaluations. It is likely that a good score in thefirst test increases the students’ self-confidence and motivation, which positively contributesto a good score in the following evaluation. Contrarily, it could be possible that studentsbecome too sure of themselves when obtaining a good mark and thereby loose ground inthe following evaluation by providing insu"cient e!ort. Including previous achievement asan explanatory variable requires both evaluation scores to be normalized. Otherwise, aninfluence between both scores might simply result from a shift in scores. Note that theformula for achievement at the entry into 3rd grade, A2 (see Figure 2), still includes thenon-cognitive skill variables. The interpretation would be that these capture the e!ect onachievement due to a time-invariant component of non-cognitive skills or ability, while theinclusion of previous achievement, A1, takes account of the evolution of non-cognitive andunobserved cognitive ability of the student.Finally, achievement at the entry into 6th grade, denoted A3, is evaluated similarly with all

prior inputs as explanatory variables. For school input, newer values of the variables replacethe older ones (I use S2 instead of S1), whereas both historical achievement measures (bothA1 and A2) are included, given that they capture di!erent types of abilities (see Section 4).

In terms of the theoretical framework of estimation strategies proposed by Todd andWolpin(2003), the modified model corresponds to a mix of value-added and cumulative specification.The specific equations used for the econometric estimation will be presented in subsection 3.3.Before that, I will introduce the detailed Oaxaca-Blinder decomposition and its assumptions,on which the empirical part of this paper will be based.

3.2 The detailed Oaxaca-Blinder decomposition

Introduced in 1973 by Blinder (1973) and Oaxaca (1973), decomposition methods are nowwidely used in labor economics (Fortin et al. 2010). The aim of the decomposition is lessto reveal concrete behavioral relations but rather to show which characteristics contributeby how much to the observed outcome di!erence between two groups (Fortin et al. 2010,p. 2). With the help of a decomposition, it is possible to distinguish three components. Thefirst, or explained, component indicates which part of the gap between the groups is due todi!erences in explanatory variables. The unexplained part indicates the part of the outcomegap which is due to di!erences in coe"cients of each explanatory variable. Any remainingdi!erences, which in econometric terms result in a higher or lower value of the intercept inthe regression, are due to di!erences in unobservable variables or are entirely unexplainable.

Having been extensively used for studies on inequalities in the labor market, their appli-cation for studying inequalities of educational outcomes is much more recent. Sakellariou(2008) uses decomposition methods to analyze the test score gap between indigenous andnon-indigenous students in Peru. McEwan (2004) does a similar exercise for Bolivia andChile. Another example is Ammermueller (2007), who decomposes the PISA score di!erencesbetween Germany and Finland. An interesting application is proposed by Barrera-Osorio,Garcia-Moreno, Patrinos and Porta (2011), who decompose PISA test score di!erences overtime for the same country (Indonesia). While these and some other papers examine testscore gaps between countries or groups of people, the only paper to my knowledge usingdecompositions to examine the test score gap between genders is Sohn (2010). Sohn (2010)uses an aggregate quantile decomposition in order to examine the gender mathematics gapin primary school using the same data as Fryer and Levitt (2010).

15

The Oaxaca-Blinder decomposition goes as follows. Suppose an outcome – in our casea test score – can be represented according to the production function approach describedin the previous subsection. If we have two groups, A and B, with di!erent linear outcomeproduction functions, we can state the expected outcome of group A as follows:

E(Y |DA) = E(X!A + "A|DA) = E(X|DA)!A + E("A) = E(X|DA)!A (1)

where Y is the outcome, DA is an indicator for group A, X is a vector of school and familyinputs which are supposed to explain the model. "A is a random error term with mean 0 anda standard deviation of #. Note that I do not apply the expectation operator to coe"cient !,given that the latter is assumed to be group-specific and therefore simply a scalar. Accordingto this setting, the expected outcome for group B is:

E(Y |DB) = E(X|DB)!B + E("B) = E(X|DB)!B (2)

If we substract the expected outcome of group B from the expected outcome of group A, weobtain, given that E("A) = E("B) = 0:

E(Y |DA)! E(Y |DB) = E(X|DA)!A ! E(X|DB)!B

Now, we suppose that in absence of discrimination, members of group B would have thesame outcome production function as members of group A. This means that in absence ofdiscrimination, the expected outcome for members of group B would write:

E(Y |DB) = E(X|DB)!A + E("B) = E(X|DB)!A (3)

If we add and subtract this expected counterfactual outcome to the di!erence of actualexpected outcomes, we obtain:

E(Y |DA)! E(Y |DB) = E(X|DA)!A ! E(X|DB)!B + E(X|DB)!A ! E(X|DB)!A

Rearranging this expression allows us to express the gap between both groups as follows:

= [E(X|DA)! E(X|DB)]!A + E(X|DB)(!A ! !B) (4)

Finally, in order to work with our data, we replace the expectation by the sample mean andobtain:

= (XA ! XB)!A + XB(!A ! !B) (5)

where XA = 1nA

!i!AXi and XB = 1

nB

!i!B Xi with nA and nB the number of individuals

in groups A and B. Note that !A and !B are written with a hat, indicating that theyrepresent the estimated coe"cients.

The first term of this decomposed sum, (XA!XB)!A, represents the “endowment e!ect”. Itis the di!erence in outcome between groups A and B, which is due to di!erences in observablecharacteristics in X, and is thus also called the explained term. The second term of thedecomposition, XB(!A!!B), is the “structural” di!erence of the production function betweengroup A and B. It indicates how the same level of characteristics are translated into a di!erentlevel of outcome for each group. In the outcome production function it is represented bydi!erent values for the coe"cient !. If we assume that the unobserved di!erences betweengenders are immutable and strictly a consequence of being a boy or a girl, this “structural”di!erence accounts for the unexplained or discriminatory part of the gap. Otherwise, thestructural component of the gap between both groups also captures di!erences in unobservedcharacteristics (Jann 2008, p. 3).

Based on the assumption that the outcome production function is linear, we can furtherdecompose the gap into the structural as well as the endowment e!ect of each explanatory

16

variable (Fortin et al. 2010, p. 38). The detailed decomposition of the endowment e!ect canbe written as follows:

(XA ! XB)!A =K"

k=1

(XAk ! XBk)!Ak (6)

The subtle di!erence between both sides of the equation is that XA is a matrix of all averageexplanatory variables for group A, whereas XAk represents the average value of explanatoryvariable k in group A. Similarly, !A is the vector of coe"cients for group A, whereas !Ak isthe coe"cient of explanatory variable k for group A.

Accordingly, the structural part of the di!erence can be decomposed into:

XB(!A ! !B) = (!A0 ! !B0) +K"

k=1

XBk(!Ak ! !Bk) (7)

The same distinction applies between XB and XBk as well as !A and !Ak. Note that Iseparately highlight the di!erence between the intercept in the regression of each group,(!A0 ! !B0). This allows me to further distinguish two elements within the structural e!ect.On the one hand, the second term,

!Kk=1 XBk(!Ak ! !Bk), represents the di!erences in

returns to each explanatory variable, indicated by di!erences in the coe"cients !Ak and!Bk. These di!erences indicate how a particular value of the explanatory variables translatesinto di!erent outcomes according to the students’ group. On the other hand, the di!erencebetween the intercept, (!A0 ! !B0), can be considered as the residual di!erence betweenoutcome measures. It is due to di!erences in unobserved characteristics between students ofboth groups or unexplainable discrimination.

Decomposition methods are relatively easy to understand and are therefore often used with-out discussing the underlying assumptions and the exact meaning of the identified parameters(Fortin et al. 2010, p. 11). The remainder of this subsection discusses the assumptions rele-vant for the detailed decompositions carried out in this paper. The following part is greatlyinspired by Fortin et al. (2010), who provide a very detailed discussion of the assumptionsfor all types of decompositions.

1. Mutually exclusive groups. To begin with, the decomposition supposes the existenceof two mutually exclusive groups (Fortin et al. 2010, p. 12). It is not possible to carry outthe decomposition if some individuals are member of more than one group, however this isnot an issue for decompositions of the gender gap.

2. Structural form and additive linearity. Second, the decomposition requires theoutcome for both groups to depend on a structural form m(·), which depends on observable(X) and unobservable (") characteristics (Fortin et al. 2010, p. 13). Hence, the outcome Yfor individual i in group A can be written YiA = mA(Xi, "i). Accordingly, for an individualof group B we write YiB = mB(Xi, "i). This is the same assumption as the one underlyingTodd and Wolpin’s (2003) production function approach and thus consistent with the modeldescribed in the previous subsection.This structural form assumption is necessary to produce the components of the outcome

variable into which we can decompose the di!erences between both groups. Given the struc-tural form, di!erences in the outcome – in our case test scores – can be decomposed intofour components. First, there may be di!erences in the coe"cients of the function mg whichdetermine the returns to observable characteristics. If the model is linear, this means thatdi!erences in the model are di!erences in the coe"cient vector !. Second, the function mg

could di!er in coe"cients determining the returns to unobservable characteristics. Third,both groups could simply characterize themselves by di!erent observable characteristics. Fi-nally, groups might di!er in unobservable characteristics (Fortin et al. 2010, p. 14-15). In

17

order for di!erences in unobservable characteristics to be identifiable, we need to make as-sumptions about their distribution. It is often not possible to separately identify the contri-bution of observed and unobserved characteristics to the outcome. In this case, it is moreappropriate to combine both as the contribution to the outcome by observed and unobservedcharacteristics (Fortin et al. 2010, p. 17). With some additional assumptions stated below,we will refer to this combined component as the “structural di!erence”.

Being more dependent on the structure of the outcome production function, the detaileddecomposition additionally requires the assumption of additive linearity. Under additivelinearity, the outcome for individual i, member of group g, can be represented with a linearadditive function:

Yig = mg(Xi, "i) = Xi!g + "ig (8)

where g is the group (A or B) – in our case gender –, Xi are observable characteristicsand "ig represents both measurement error and the impact on the outcome of unobservablecharacteristics for individual i. The coe"cient !g indicates the impact of each observablecharacteristic in X on the outcome variable for a member of group g. Note that by statingthis model we also assume that the coe"cient !g is specific for each group, meaning that weallow for the observable characteristics to have a di!erent impact on boys than on girls. Fortinet al. (2010, p. 27) point out that the additive linearity assumption could be omitted if oneassumes that for both groups, observable characteristics are independent from unobservablecharacteristics. Moreover, one would have to assume that " is a scalar random variable andthat mg(X, ") is strictly increasing in ". With both assumptions, it would be possible tonon-parametrically identify the structural functions producing the outcome of interest.

3. Simple counterfactual treatment. Decomposing the gender gap in test scores re-quires the existence of a counterfactual outcome describing the theoretical score in absenceof discrimination. We assume a simple counterfactual determination of the outcome. Thismeans that the counterfactual outcome, which a member of group A would achieve if she werein group B, is the result obtained by applying the outcome production function of group Bwith the characteristics of the member of group A. With other words, for the theoretical testscore of a girl in absence of a gender gap we assume the outcome which a person of this girl’scharacteristics would obtain if her scores were determined by the boys’ outcome productionfunction. Alternatively, we can take the girls’ outcome function as the baseline and definethe counterfactual outcome for boys in absence of discrimination as the outcome that a girlwith the boy’s characteristics would have obtained.The group whose outcome is used to determine the counterfactual outcome in absence of

discrimination is called the “reference group”. As the magnitude of the components obtainedin a decomposition is evaluated with the characteristics of the reference group, the choiceof the latter influences the result obtained. Therefore, the choice of the reference groupmust be well justified. We need to ask whether it is more realistic that in absence of genderdi!erences girls would obtain the scores that boys’ test score functions would predict for girls’characteristics or whether boys would obtain the scores that girls’ test score functions wouldpredict for boys’ characteristics. This question is, of course, purely hypothetical and cannotbe answered with logical reasoning. Usually, research on gender di!erences uses males as thereference group (Fortin et al. 2010, p. 38). Given that an objective answer does not exist andin order for my results to be consistent with previous research, I will therefore treat boys asthe reference group. In order to be as clear as possible, all test scores of boys were normalizedto have a mean of 0 and a standard deviation of 1.We could also realize the decomposition in a more complicated way by assuming di!erent

counterfactual outcomes (Fortin et al. 2010, p. 14). For example, we could assume that inabsence of discrimination, the score of both boys and girls would be determined by a thirdfunction. Indeed, Fortin et al. (2010, p. 44) suggest to consider a weighted average of both

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groups’ production functions as the counterfactual production function without discrimina-tion. Alternatively, the coe"cients obtained from a regression with both groups pooled canbe used as the counterfactual. This is similar to performing a regression with the pooledsample including a group dummy as explanatory variable. In this dummy variable approach,the coe"cient on the group dummy can be considered as the structural component of thegroup di!erence.

4. Conditional independence, ignorability or zero conditional mean. The mostimportant assumption required to distinguish between the components of the outcome gapis the ignorability or conditional independence assumption. We need to assume that theerror term ", which accounts for measurement errors as well as unobserved characteristics, isindependent of the individual’s group at all possible values each of the explanatory variablescan take. Formally stated, if Dg is the indicator stating whether an individual belongsto group g (where g can be A or B), " is assumed to be independent of Dg given for allpossible values of each X, or Dg "" "|X for all X = x. This assumption is equivalentto the unconfoundedness or selection on observables assumption in the program evaluationliterature (Fortin et al. 2010, p. 20). In terms of the decomposition components describedabove, the ignorability assumption states that we correctly identify the part of the gap whichis due to di!erences in observable characteristics without confounding it with unobservablecharacteristics.)

The detailed decomposition requires a more specific formulation of the conditional in-dependence assumption, which is extensively used under the name zero conditional meanassumption. It states that the error term "ig is conditionally independent from Xig and fromthe group g. Formally,

E(vg|X,Dg) = 0 (9)

The zero conditional mean assumption was used in the transformation of equations (1) and (2)and allows us to rewrite the observed and counterfactual outcomes for both groups as follows:

Group A, observed outcome: E(Y |DA) = E(X|DA)!A (10)

Group A, counterfactual outcome: E(X|DA)!B (11)

Group B, observed outcome: E(Y |DB) = E(X|DB)!B (12)

Group B, counterfactual outcome: E(X|DB)!A (13)

The formulations for the observed and counterfactual outcome of each group provided inequations 10 to 13 allow us to decompose the gender gap as described above. The zeroconditional mean assumption is therefore an important assumption for the decomposition tobe valid.Fortin et al. (2010, p. 5) point out that in order to detect a causal e!ect, the zero conditional

mean assumption can be replaced by an ignorability assumption, stating that the distributionof the error term is the same for any possible value of x. In this case, the decomposition wouldeven be valid in the presence of selection biases, under the condition that the selection biasis the same in both groups.There are three cases in which the conditional independence assumption may not hold

(Fortin et al. 2010, p. 23). First, members of both groups might select themselves into theactivity, for which the gap is examined, according to their background characteristics. Thisis a serious concern in research on the gender gap such as the one by Ellison and Swanson(2010), who examine gender di!erences among students who choose to participate in veryadvanced mathematics competitions. In their paper, selection is a concern, as girls might

19

not participate in the competition for the simple fact that they are reluctant to competition(Niederle and Vesterlund 2010). In our case, selection into the activity would be problematicif boys and girls could choose whether they enter into school or not. This is, of course, notthe case for primary school, which is mandatory for every child. Moreover, participants inthe French Primary School Panel were chosen at random.

Second, individuals might select themselves into the two groups according to their char-acteristics. This is the case in applications where it is possible to choose the group, such asresearch on the wage gap between members and non-members of a union. This case is of norelevance for an analysis of the gender gap.

The third case is more relevant for our topic. The conditional independence assumptiondoes not hold if observed and unobserved characteristics (X and ") are correlated. Giventhat educational achievement is likely to depend on unobservable characteristics which arerelated to observable ones such as the family background, the implications of the failure ofthe conditional independence assumption need to be discussed. I address this issue in thefollowing way.

On the one hand, if we can assume that the correlation between observed and unobservedcharacteristics is the same for both groups, the aggregate decomposition remains valid (Fortinet al. 2010, p. 87). If, for example, the child’s ability is correlated with the education of hisparents and both determine the child’s achievement, the Oaxaca-Blinder decomposition is notable to correctly identify the part of the gender gap which is due to di!erences in parent’seducation. However, if we assume the correlation between parents’ education and the abilityof the child to be the same for boys and girls, the decomposition can still provide a meaningfulinterpretation, given that the bias is the same for both genders. The assumption of identicalcorrelations between observable and unobservable characteristics for boys and girls can bedefended as reasonable, because we have no explanation on why this should not be thecase. At the same time, we are willing to assume that the coe"cients in the achievementproduction function di!er between boys and girls. If we accept this, how can we be sureenough to assume that the correlation between observable and unobservable characteristicsis the same for both genders? Unfortunately, this question will remain unanswered, giventhat unobserved characteristics cannot be measured.

On the other hand, I try to minimize the influence of non-observable characteristics onachievement by making use of the large number of variables available in the French PrimarySchool Panel. In particular the availability of an evaluation of each child’s non-cognitiveskills leads me to assume that once these non-cognitive skills are included as explanatoryvariables in the achievement production function, any remaining unobservable characteristicsare independently distributed and therefore satisfy the conditional independence assumption.

5. Invariance of conditional distribution. In order for the counterfactual achievement– the achievement we would observe in absence of discrimination – to be valid, we mustassume that the conditional distribution of unobserved characteristics (") given observedcharacteristics (X) remains valid if the function determining the outcome is changed. Inother words, we suppose that the outcome which is obtained by plugging girls’ characteristicsinto the boy’s outcome function is a valid counterfactual outcome for girls in absence of genderdi!erences. This assumption rules out general equilibrium e!ects. Fortin et al. (2010, p. 22)point out that the invariance of conditional distribution assumption holds if the assumptionof a simple counterfactual treatment as well as the conditional independence assumptionare satisfied. With other words, the simple counterfactual assumption does not hold whenwe consider the existence of general equilibrium e!ects (Fortin et al. 2010, p. 15). Hence,decomposition methods rely on the assumption of a partial equilibrium framework. Theexplainable and unexplainable components of an observed gap between two groups mightdi!er if a policy is implemented for the entire population.

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6. Overlapping support. A final and merely technical assumption necessary to implementdecomposition methods is the existence of overlapping support. We need to assume that nosingle value of observable or unobservable characteristics allows to determine an individual’sgroup. This is, of course, true in our case, where none of the background characteristicscan be ruled out for either boys or girls. The situation is less clear for unobservable char-acteristics, because we might think of genetical or educational reasons for which one genderhas certain characteristics which the other does not. If the overlapping support assumptionholds, we know that the three components of the gender gap initially described – di!erencesin endowments, di!erences in returns to endowments and unexplainable di!erences – entirelycapture di!erences in outcome between both groups.

Jann’s (2008) command oaxaca is very comfortable for implementing the Oaxaca-Blinderdecomposition with Stata. It has many features allowing all types of aggregate and detaileddecompositions. Concerning the choice of the reference group, the command allows to useeither group as the reference, but also to define the counterfactual as a weighted averageof both groups’ outcome production function or as the production function obtained from apooled regression.Jann (2008, p. 6) highlights an important point about the variance. Given that not only

the outcome (in our case the test score), but also explanatory variables are subject to mea-surement errors, the latter also have a sampling variance and therefore must be considered asrandom variables. While standard regressions can be carried out no matter whether explana-tory variables are stochastic or fixed, this is not the case for the Blinder-Oaxaca decompositionbecause we need to multiply coe"cients with means of explanatory variables. Jann’s (2008)Stata command oaxaca provides consistent standard errors.

In order to account for the fact that unobserved heterogeneity might be clustered at theschool-level, Sakellariou (2008) uses Feasible Generalized Least Squares in order to accountfor standard deviations depending on the school level. For my study, three methods seempossible: use school fixed e!ects, cluster the standard errors according to school or including aset of school-specific covariates. As the Primary School Panel contains only between 5 and 10students per school, school fixed e!ects would considerably increase the number of variables.Therefore, I use the school-specific characteristics contained in the Primary School Panel inorder to account for influences from the school. Please refer to Section 4 for a description ofthe relevant variables.

While the algebraic reasoning for the aggregate and detailed decomposition is simple tounderstand, two issues have to be highlighted in the case of categorical explanatory variables,which I intensively use. For these variables, the results vary according to the choice of theomitted category. Such variations are due to the fact that the contribution of each includedcategory to the endowment e!ect of the gender gap are expressed in terms of the di!erencewith respect to the omitted category (Fortin et al. 2010, p. 40). In spite of di!erences ineach category’s e!ect according to the choice of the omitted category, the choice of referencecategory does not influence the overall e!ect of all categories. The overall e!ect remainsunchanged, because it is the sum of the endowment e!ects for each category.

Second, it is not possible to distinguish between-group di!erences of the intercept fromdi!erences of the coe"cient for the omitted category (Jones 1983, Oaxaca and Ransom 1999).As Fortin et al. (2010, p. 41) point out, the decomposition of the structural (unexplained)gap between both groups of a categorical variable with m possible values can be rewritten asfollows:

XB(!A ! !B) = [(!A0 + !A,cat1)! (!B0 + !B,cat1)] +"

m "=1

XB,catm(!A,catm ! !B,catm) (14)

if category 1 is omitted and

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XB(!A ! !B) = [(!A0 + !A,cat3)! (!B0 + !B,cat3)] +"

m "=3

XB,catm(!A,catm ! !B,catm) (15)

if category 3 is omitted. In both cases, !A0 (!B0) cannot be distinguished from !A,cat1 nor!A,cat3 (!B,cat1 nor !B,cat3) respectively. While the aggregate structural component of thegender gap is the same in both cases (because we do not distinguish between the intercept andthe variable coe"cients), we obtain di!erent contributions of the structural e!ect of variablecat on the output gap, depending on the category we choose to omit.

Some authors, like Yun (2008), propose solutions based on constrained linear regressions,in which the coe"cient of the omitted category is constrained to be a weighted average ofthe coe"cients of the other categories. Horrace and Oaxaca (2001) propose to separatelyestimate the contribution to the gap for each category and then calculate their share of thetotal impact of that variable. However, Fortin et al. (2010, p. 42) point out that a clearsolution does not exist. They state that such modified estimators are di"cult to interpretand diminish the understandability of the detailed decomposition.With binary explanatory variables, it is still not possible to separately identify the intercept

from the coe"cient of the omitted category. However, the absolute value of the coe"cientwill be the same no matter which of the two categories is chosen as reference. In order notto have to cope with the reference category problem, I will thus transform all categoricalexplanatory variables into binary variables.Another point highlighted by Jann (2008) is related to the omitted category issue discussed

above. In the detailed decomposition, the unexplained part of each variable is sensitive toscaling. For example, if all values for both groups of a variable are shifted by a, the share ofthe constant in the unexplained part of the decomposition increases. Therefore, the detaileddecomposition only makes sense for variables which have a natural scale. In this paper, allvariables take one of the following three scales. First, most variables describing family andschool inputs are binary. Second, some input variables are discrete and indicate the number ofstudents in a class or the number of brothers and sisters. Finally, test scores were normalizedsuch that boys score with a mean of 0 and a standard deviation of 1. Hence, the scalingproblem is not an issue in the analysis of this paper.This paper aims to examine the components of the gender gap in mathematics using

the detailed Oaxaca-Blinder decomposition. As presented in this subsection, the detaileddecomposition is based on a number of hypotheses which are not always easily acceptable.Nonetheless, I consider this method as the most appropriate for our goal. Two other methodspursue a similar aim and are preferred by other authors. However, they do not fulfill thesame purpose.First, Elder, Goddeeris and Haider (2010) presents the so-called “dummy-variable method”

as an alternative to the Oaxaca-Blinder decomposition. Using the entire sample of bothgroups, the dummy-variable method complements the usual explanatory variables by includ-ing a group dummy as well as interaction terms of the group dummy with all explanatoryvariables. Now, the coe"cients of the explanatory variables without the interaction termcorrespond to the explained part of the between-group di!erence. Moreover, the coe"cientsof the explanatory variables interacted with the group dummy correspond to the di!erence inreturns for an input. The entirely unexplainable part of the gap is captured by the coe"cientof the group dummy without interaction after having controlled for all possible explana-tions. While this method attracts by its simplicity, it is impractical with a large number ofexplanatory variables as we use in this paper.Second, decomposition methods have been developed for decompositions that go beyond the

mean. Sohn (2010) is, to my knowledge, the first to use them in order to analyze the factorsthat contribute to the gender gap in mathematics in primary school using US data. Working

22

with the same data as Fryer and Levitt (2010), he uses the quantile version of the aggregatedecomposition proposed by Melly (2006) and finds that the gender gap is almost entirelydue to the structural component of the decomposition. This is not surprising as backgroundcharacteristics of boys and girls are almost identical. It is not possible to understand theemergence of the gender gap in mathematics without examining the contribution of di!erenttypes of input. An aggregate decomposition is thus not enough to obtain interesting results.However, using a quantile version of the detailed decomposition requires strong additionalassumptions concerning the counterfactuals at each quantile of the distribution (Fortin etal. 2010, p. 72). Moreover, the results become very complicated to interpret. In my view,the added value of performing the detailed decomposition for the entire distribution is notjustified by the limitations imposed by additional assumptions. Therefore, I restrict myinterest to the decomposition at the mean.

3.3 The empirical model

Having discussed the structure and the assumptions of the model and the decompositionin detail, I will now present the empirically estimated model equations. The estimationequations presented here are based on the model illustrated in Figure 2. Working with areduced form of this model, we need to keep in mind the correlations between di!erent typesof inputs when interpreting the results. The reduced form version of this model for eachindividual can be written with four equations:

A1i = a0 + $ia1 + F #0ia2 + "1i

A2i = b0 + $ib1 + F #0ib2 + F #

jk1ib3 + F #l1ib4 +A#

1ib5 + S#1ib6 + "2i

A3i = c0 + $ic1 + F #0ic2 + F #

jk1ic3 + F #l1ic4 +A#

1ic5 +A#2ic6 + S#

2ic7 + "3i$i = d0 +NC #

id1 + %i

(16)

All individual-specific variables have the subscript i. A1i to A3i are the achievement measuresin 1st, 3rd and 6th grade (CP, CE2 and sixieme). As described in Figure 2 above, F0i

comprises family and maternal school inputs, which were determined before the entry intoprimary school and are thus the only variables explaining achievement at the entry into school,A1i. Family input after the entry into primary school is denoted on the one hand with Fl1i,describing the input types which are a!ected by the child’s school achievement, such as thecontact with as well as the parents’ opinion about school. On the other hand, family inputdetermined at home, such as the child’s afternoon activities, is denoted as Fjk1i. School-specific characteristics are included in vectors S1i and S2i. Note that only the most recentschool input is included in the achievement production functions for A2i and A3i, which isconsistent with the model described in Figure 2. "1i, "2i and "3i capture measurement errorsand the impact of unobservable characteristics other than the child’s ability on achievement.$i represents the individual-specific characteristic, which is often described as each stu-

dent’s ability. Of course, it is not possible to observe the ability of each individual. Thereforewe make the assumption that ability can be approximated with the evaluation of non-cognitiveskills included in the French Primary School Panel. We must keep in mind that this measureis both a subjective assessment of the student’s non-cognitive skills and an imperfect sub-stitute for her or his actual ability. For simplicity, we assume a linear relation between thenon-cognitive skills evaluation, denoted NCi, and the ability measure $i. d0 is the interceptand can be interpreted as a scaling factor. %i captures measurement errors as well as theimprecision related to the use of non-cognitive skills as an approximation for ability.

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Given that we do not observe $i directly, we must substitute it into each of the achievementfunctions. Hence, the actually estimated equations are as follows:

A1i = [a0 + d0a1] +NC #id1a1 +F #

0ia2 +[%ia1 + "1i]

A2i = [b0 + d0b1] +NC #id1b1 +F #

0ib2 + F #jk1ib3 + F #

l1ib4 +A#1ib4 + S#

1ib5 +[%ib1 + "2i]

A3i = [c0 + d0c1]# $% & +NC #id1c1# $% & +F #

0ic2 + F #jk1ic3 + F #

l1ic4 +A#1ic4 +A#

2ic7 + S#2ic8# $% &

+ [%ic1 + "3i]# $% &

intercept ability explanatory variables error term

Again, the equations are expressed for each individual and contain the subscript i for allindividual-specific variables. Writing the individual-specific equation facilitates the distinc-tion between individual-specific variables and structural parameters. For better readability,the terms are grouped according to their role. The first two underlined terms correspondto the intercept and the non-cognitive skills variable which aims to approximate individualability. The remaining part of each equation contains all explanatory variables (family andschool input as well as prior achievement). Each equation ends with the error term.

Having substituted the separate individual ability equation into each achievement equationallows us to be more clear about the parameters we estimate. As the above equations suggest,the intercept is a composite term containing the intercept of the achievement equation (a0,b0 and c0 respectively) as well as the intercept of the ability equation, d0, weighted by theability’s impact on each of the achievement measures (a1, b1 and c1). If the decompositionreveals structural di!erences in the intercept between boys and girls, these are likely to becorrelated with structural gender di!erences in the returns to non-cognitive skills (meaninggender di!erences in d1a1, d1b1 or d1c1). A similar correlation may be true for the errorterm. In the same way as the intercept, the error term is a composite between the error termof the achievement equations ("1i, "2i and "3i) and the error term of the ability equation,%i, weighted by the ability’s impact on each achievement measure (a1, b1 and c1). Thus,the error term is correlated with the coe"cient relating non-cognitive skills to achievement(d1a1, d1b1 and d1c1 respectively) if the error in the ability equation, %i, is not expected tobe 0. This would be the case if the equation relating non-cognitive skills to ability would beendogenous.To sum up, the model provides us with consistent estimates of the components of the

gender gap if we consider the following assumptions as reasonable. First, the assessment ofnon-cognitive skills is a good proxy for the child’s inherent ability. Any measurement error oromitted variables, %i, are not correlated with ability, $i. Second, after including non-cognitiveskills as a proxy for ability into the achievement equation, the error term of the achievementequations, which captures measurement error and omitted variables, is not correlated withthe student’s ability. With other words, "1i, "2i and "3i respectively, are not correlated withd0, the intercept of the ability equation. Third, the potential correlation between the errorterm of the achievement equations, "1i, "2i and "3i respectively, and any explanatory variableis the same for boys and girls. And finally, non-cognitive skills are constant through time.Given that the three achievement equations partly depend on the same explanatory vari-

ables, it is optimal to estimate them using seemingly unrelated regression methods. Seeminglyunrelated regression models are estimated with generalized least squares. There are somecases in which GLS is more e"cient than OLS (Greene 2008). However, in order to keepthe analysis and its interpretation as simple as possible, the Oaxaca-Blinder decompositionis carried out with OLS. In order to test the robustness of our results, the regression modelsunderlying the decomposition will also be carried out as seemingly unrelated regressions.

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4 The data

This paper uses data from the French primary school panel (“Panel de premier degre”),which contains information about the school career of approximately 10,000 randomly selectedstudents who entered primary school in France in 1997. France’s Ministry of education(Ministere de l’Education Nationale, de la Jeunesse et de la Vie associative) follows thesestudents until the end of their school career. For each school year, administrative informationabout the child’s school environment is collected: number of students, electives chosen as wellas general information about the school. Students were even followed when they changedclasses or when they had to repeat a class.

In addition to administrative school data, a parent questionnaire was administered in thesecond school year. A large variety of questions was asked aiming to obtain informationabout the parents’ background characteristics, the child’s maternal school history as wellas its current free-time activities and other facts allowing to assess the parents’ educationalphilosophy. Moreover, questions assessing the parents’ opinion about a number of school-related issues allow to approximate unobservable background characteristics. In order tocomplete the information collected in the parent questionnaire, the school principal wasasked to provide the basic facts on socio-economic background, such as family compositionand the socio-professional category of both parents.

Finally, and most importantly, three standardized tests assess the students’ cognitive abil-ity, in particular in mathematics and French. In order to evaluate these skills regularly, thenational evaluations at the entry into 3rd and into 6th grade were complete by an additionalstandardized test at the entry into primary school, which was created explicitly for the Pri-mary School Panel. It evaluates the students’ most basic skills in general knowledge, Frenchand mathematics. The evaluation at the entry into primary school was completed with anassessment of each students’ non-cognitive skills by the class teacher.

Table 2 gives an overview of the five years of French primary school as well as the dates atwhich information was collected. Primary school in France starts with the so-called “Classepreparatoire”, which is followed by two years of elementary courses (“Cours elementaire”) aswell as two years of intermediate courses (“Cours moyen”). After successful completion ofthe five primary school years, students enter into middle school (the “college”), where theyspend four years before entering high school. The first year of middle school is called 6thgrade (“sixieme”).

As can be seen in Table 2, the abovementioned standardized evaluations took place at theentry into 1st grade (“classe preparatoire”), at the entry into 3rd grade (“cours elementaire2”) and at the entry into 6th grade (“sixieme”). Parent questionnaires were only administeredin 2nd grade, which means that their answers might be influenced by the experience made inthe first primary school year.

Table 2: Data schedule of the French Primary School Panel 1997

Year School type Class Evaluation/Data collected

1997-1998 CP (Classe preparatoire) Evaluation mathematics, Frenchgeneral skills, non-cognitive skills

1998-1999 Ecole Primaire CE1 (Cours elementaire 1) Parent questionnaire1999-2000 (Primary school) CE2 (Cours elementaire 2) Evaluation mathematics, French2000-2001 CM1 (Cours moyen 1)2001-2002 CM2 (Cours moyen 2)2002-2003 College 6th grade (sixieme) Evaluation mathematics, French

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Table 3: Overview of the cognitive skills evaluated at the entry into primary school

Skill type Skill

General skills General knowledgeKnowledge of spatial conceptsKnowledge of temporal conceptsTechnical knowledge

French Familiarity with writingVerbal knowledgeReadingOral comprehension

Mathematics NumerationNumbers & geometry

The remainder of this section presents the details on the standardized tests as well assummary statistics for the background characteristics used to examine the determinants ofthe gender gap in mathematics. While I provide detailed summary statistics for backgroundcharacteristics in this section, the presentation of the standardized tests will be limited toa description of the tasks composing the evaluation, given that Section 5 will be entirelydevoted to a detailed quantitative description of the gender gap.

4.1 Standardized tests

Evaluation of cognitive skills at the entry into primary school2

The general test at the entry into primary school aims to assess the students’ skills concerninggeneral knowledge, verbal knowledge, familiarity with writing, logical skills, familiarity withnumbers as well as their mastering of concepts related to time and space. Test results wereeither based on written exercises or on observations by teachers. The tests were chosen tocorrespond to concepts and skills children learn at the age of six years (Colmant et al. 2002).Thus, they were designed with the aim to test children on some concepts which a normallydeveloped child might not yet be familiar with. Table 3 provides an overview about theskills tested. In order to be able to interpret this paper’s results, the details of the tests areprovided in the following.

General skills

1. General knowledge. Students are asked whether they know typical stories, comics orinstruments. They have to circle the object named by the teacher among the pictureson their answering sheet.

2. Knowledge of spatial concepts. This test assesses whether students are familiar withnotions such as “up” vs. “down”, “above” vs. “below”, “in front of” vs. “behind”, “left”vs. “right” as well as “between”. On their answering sheet, children have to mark thesymbol indicated by the teacher, for example “the animal to the left of the dog”.

3. Knowledge of temporal concepts. Here, it is evaluated whether children know conceptssuch as “beginning” vs. “end”, “before” vs. “after”, “momentarily”, “start” vs. “finish”as well as the fours seasons. In order to assess the knowledge of these concepts, childrenhave to finish a story or identify objects on a sheet of paper. Colmant et al. (2002) ob-serve that the ability to correctly complete a short story depends on having understood

2A detailed description of the tests can be found in Colmant, Jeantheau and Murat (2002)

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the notions “beginning” and “end” as well as “before” and “after”. Moreover, Colmantet al. (2002) highlight that the acquisition of spatial skills precedes the acquisition oftemporal skills, given that the results observed for the latter are much weaker.

4. Technical knowledge. Unfortunately, it was not possible to evaluate practical skills suchas constructing or repairing something. Also, the conception of a test evaluating thetechnical knowledge could not be based on existing examples, as little research aboutstrategies to evaluate these skills at the beginning of primary school exists. Therefore,an external validity for the scores cannot be guaranteed (Colmant et al. 2002, p. 65).Questions were chosen based on the results of a pre-tested series of questions. Childrenwere asked to complete the following tasks: 1) On a sheet with several pictures ofobjects, identify those that measure the time. 2) Among pictures of four light bulbs,identify the one that fits in the depicted pocket lamp. 3) Among objects, identify thosethat cut. 4) Identify and color an object on a depicted machine (e.g. the tyre of a car).5) Match two vehicles according to “where” they drive (on the water, in the air, ...).6) Identify the false object in a list of objects which are used in a particular profession.

French

1. Familiarity with writing. This test asks whether children can identify what a text is,whether they know the alphabet etc.

2. Verbal knowledge. Children were asked to write or identify words, write letters, memo-rize series of pluses and minuses as well as memorize series of letters.

3. Reading. The evaluation of children’s reading ability at the entry into primary schooluses the fact that knowing the alphabet is a good predictor of being able to read(Colmant et al. 2002). In order to test their knowledge of reading, children were askedto complete the following tasks: 1) Identify an orally given word among four writtenwords. 2) Identify an orally given, non-existing word (pseudo-word, e.g. “mida”)among four written pseudo-words. 3) Identify letters (e.g. which has the name “ef”).4) Identify letters (e.g. which describes the sound “n”). 5) Identify the word whichdoes not start, end or sound the same as the others. 6) Identify the new word that iscreated by taking away the first letter(s) of a word. 7) Judge whether a sentence is incorrect grammatical form. 8) Judge whether a sentence is false because the the wordsare in disorder or whether a word is wrongly spelled.

4. Oral comprehension. When a teacher tells a story to his students, it is not easy to testthe capacity to understand the words, as the child might understand some of the storyfrom the teacher’s mimic or from previous knowledge. Moreover, if the test consists inasking questions about a previously told story, it is also not easy distinguish betweentesting the capacity to understand and the capacity to memorize. Therefore, the testconsists of highlighting the object among pictures on the answer sheet that “fits” tothe story told by the teacher.

Mathematics

1. Numbers and geometry. The first of two tests evaluating the mathematical ability ofchildren at the entry into primary school aims to assess whether children know numbersand geometric figures. The test is based on the theory that children independently learnthree ways of representing numbers: The written word (e.g. “seven”), the number (e.g.“7”) and the spoken word (e.g. “se-ven”). The link between these representations is notself-evident, as each follows its own logic. The third consists of a distinct word for eachnumber. On the contrary, the first representation, the written word, consists of three

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types of expressions: “one” to “nine”, “ten” to “ninety” and the special cases “eleven”to “nineteen” (“onze” to “seize” in French). These as well as the multipliers “-hundred”,“thousand” and “million” are combined using either the summation rule (“twenty-four”= twenty plus four) or – a special case of the French language – the multiplying rule(“quatre vingt” = four times twenty). Conversely, the second representation, the arabicway of writing numbers, merely consists of the ten digits “0” to “9”, which, accordingto their position in the number have a particular value.

The children’s knowledge of these representations of numbers consists of correct identi-fication of numbers in several formats. Moreover, a similar test assesses the knowledgeof geometrical figures. According to Colmant et al. (2002), the results of these tests areparticularly correlated with the results of the test in numeration, writing and knowledgeof temporal concepts.

2. Numeration. After having evaluated the knowledge of numbers, children are tested onwhether they are able to work with them. They are asked to complete the followingtasks: 1) Indicate the subsequent number in a series. Unsurprisingly, children succeedless when numbers get higher. 2) Compare quantities in order to test whether the childhas a sense of cardinality. For example, the child has to indicate in which of two barsthere are more points, the bars having di!erent lengths or di!erent densities in points.3) Solve simple arithmetic problems (2 additions, one subtraction). 4) Count.

Evaluation of skills in mathematics and French at the entry into 3rd grade

The standardized evaluation at the entry into 3rd grade (CE2) is not specific to the primaryschool panel and has to be taken by all children in French primary schools. The test wascreated in 1989 with the aim to help teachers judge strengths and weaknesses of the studentsin their class. Moreover, they allow national comparisons of students between schools. Thefollowing details concern the test in 1999, which was taken by 75.3 percent of all students inthe sample. A description of the tests can be found in Colmant, Dupe and Robin (1999).

Table 4 gives an overview of the competences evaluated at the entry into CE2. The test inmathematics consists of four components:

1. Numerical exercises. The test evaluating children’s numeration ability consists of ques-tions concerning mental arithmetic (9 questions), written summation (7 questions) andwritten subtraction and multiplication (5 questions). Moreover, children were asked totransform numbers into words and inversely, order numbers, compare written numberswith the results of simple computations.

2. Geometry. This part of the test evaluates whether children are able to orientate them-selves in coordinate system, draw a geometric figure, draw a square around some given

Table 4: Overview of the cognitive skills evaluated at the entry into CE2 (3rd grade)

Skill type Skill

Mathematics Numerical exercisesGeometryMeasuringProblem solving

French ComprehensionLanguage toolsWriting

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symbols (an exercise which seems to be more di"cult), draw the mirror image of agiven figure with the help of a pattern. Moreover, they are asked to identify one cardamong four with a pattern of geometric figures according to a short description (testsamong others spatial skills), identify information on a floor plan, answer questions ona 3-dimensional representation of a street situation (spatial skills) and trace a squareas well as a rectangle by combining given points.

3. Measuring. In addition to numerical exercises and geometry, children were tested ontheir capacity to measure time, distances and sizes. They were asked to identify in-formation on a time schedule and a clock, to use a ruler to measure the length of aline as well as draw a line of a given length. Moreover, the test consisted of questionsrequiring to sort lines according to their length, identify the correct unit from textualcontext, identify information on a calendar and compare distances. Finally, studentshad to solve problems requiring to evaluate the size of objects (lenght, width, depth)and choose the correct unit according to the context (second, minute or hour).

4. Problem solving. Finally, in addition to the evaluation at the entry into primary school,the mathematics evaluation in 3rd grade contains a part on problem solving. Studentswere asked to complete a table with information given to them, identify informationin a document and solve short problems requiring addition or subtraction. Moreover,they had to solve short problems requiring multiplication as well as the comparison ofnumbers.

Evaluation of skills in mathematics and French at the entry into 6th grade

Table 5: Overview of the cognitive skills evaluated at the entry into sixieme (6th grade)

Field Skill type Skill

Mathematics Knowledge of topics Numerical exercisesGeometryWriting numbersTreating operatorsTreating information

Knowledge of methodology Analyzing a situationApplying a techniqueProducing an answerResearching informationUsing knowledge

French ComprehensionLanguage toolsWriting a textSpelling wordsKnowing to read

The third standardized evaluation takes place at the beginning of 6th grade (sixieme),which is the first year in middle school (college) after having spent five years in primaryschool. Thus, the evaluation aims to provide a detailed overview of the cognitive skills inFrench and mathematics acquired throughout primary school. Table 5 provides an overviewof the evaluated concepts. The mathematics evaluation consists of 39 questions with 79 sub-questions in total, each of which tests at the same time the knowledge in one of five topicsas well as the knowledge of one of five methodologies (Dauphin, Rebmeister and Zelty 2003).

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The topics tested are: Numerical exercises, geometry, writing numbers, treating operatorsand treating information. Moreover, the knowledge of the following methodologies was tested:Analyzing a situation, applying a technique, producing an answer, researching informationand using knowledge.

4.2 Non-cognitive skills

Table 6: Summary statistics and t-test for non-cognitive skills

Boys Girls t-test of di!erencesSelf-confident during school activities 0.639 0.662 -0.023$ (0.012)Never fails due to excessive confidence 0.599 0.679 -0.080$$$ (0.012)Capable of regular attention 0.629 0.726 -0.097$$$ (0.012)Actively participates in group activities 0.710 0.784 -0.074$$$ (0.011)E"ciently completes tasks 0.673 0.745 -0.072$$$ (0.011)Autonomous 0.645 0.719 -0.074$$$ (0.012)Never fatigues during school activities 0.605 0.626 -0.021$ (0.012)Consciously intervenes in class discussions 0.696 0.722 -0.026$$ (0.011)Anticipates and is organized 0.575 0.653 -0.078$$$ (0.012)Good linguistic level compared to class average 0.630 0.683 -0.053$$$ (0.012)Requires support and encouragement 0.326 0.323 0.003 (0.012)Requires corrective measures and warning 0.203 0.090 0.112$$$ (0.009)Rapidly completes tasks 0.626 0.685 -0.058$$$ (0.012)Has no di"culty in activities involving gestures 0.680 0.778 -0.097$$$ (0.011)Actively participates in class discussions 0.696 0.686 0.010 (0.012)Integrates herself/himself well in class 0.630 0.684 -0.054$$$ (0.012)Observations 3258 3164(in percent) 50.7% 49.3%

Columns: mean value for boys and girls, mean and standard deviation of di!erence between boys and girls.

All variables are binary, means thus indicate the share of students for whom the item is true.! p < 0.1, !! p < 0.05, !!! p < 0.01

In addition to the standardized test at the entry into primary school, teachers were asked toanswer questions on each student assessing non-cognitive abilities such as their participationin class and their interaction with other students. They were asked to do so shortly after thebeginning of the first year of primary school, thus the evaluation aims to indicate the level ofnon-cognitive skills with which the students enters into school. As proposed in Section 3, theevaluation of non-cognitive skills can therefore be used as a proxy for each student’s abilityor individual-specific e!ect.

Table 6 shows the summary statistics for all non-cognitive skill evaluations by gender. Themeans are shown for boys and girls of the subsample of students used for the decompositionof the gender gap at the entry into primary school (see Section 6). The subsample consists of6422 students, equally subdivided into boys and girls. Moreover, the di!erence between bothmeans as well as the standard deviation of the di!erences is shown in the third and fourthcolumn of the table. Stars indicate significance levels of the di!erences. In order to avoid thereference group problem highlighted in Section 3, all variables are binary, the mean valuesthus correspond to the share of boys and girls for whom the respective statement is true. Thenon-cognitive skills above the horizontal line correspond to those that turn out to be relevantfor explaining achievement and which are thus included in all decompositions. The variablesfiguring below the horizontal line were omitted in the decompositions presented in Section 6.

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All variables except the requirement of support or corrective measures indicate a positivenon-cognitive skill. We can see that teachers issue a positive evaluation to approximately 70%of all students for each of these evaluations. Nonetheless, girls are systematically evaluatedbetter than boys. The requirement of support and encouragement as well as the activeparticipation in class seem to be the only evaluations which do not significantly di!er betweengenders. For all other skills, girls are between 2 and 11% more likely to obtain a positiveevaluation.

Gender di!erences in the evaluation of non-cognitive skills follow a stereotypical pattern.Girls are particularly more likely to never fail due to excessive confidence, be capable ofregular attention, to have no di"culty in activities involving gestures, to anticipate and tobe organized. For each of these skills, girls are almost 10% more likely to obtain a positiveassessment. Also, teachers are more than 11% more likely to consider that boys requirecorrective measures and warnings than girls. For active participation in group activities,e"cient as well as rapid completion of tasks, autonomy, linguistic level and integration intoclass, girls are more than 5% more likely to obtain a good score. Moreover, girls have a smalladvantage with respect to their evaluation of self-confidence, fatigue, as well as consciousintervention in class discussions.

4.3 Background characteristics, family and school input

Background characteristics are available for all students concerning their family environment,maternal school career as well as primary school input. In order to be coherent with themodel described in Section 3 and to be readable relative to the results presented in Section 6,background characteristics will be grouped into the same input types for the data descriptionin the remainder of this section.

Table 7: Summary statistics and t-test for family inputs prior to primary school

Boys Girls t-test of di!erencesNumber of children in family 2.582 2.626 -0.044 (0.030)Mother works 0.672 0.665 0.007 (0.012)Father works 0.930 0.934 -0.004 (0.006)Socio-professional category 3 or 4 (1997, father) 0.356 0.368 -0.012 (0.012)Mother has obtained university diploma 0.246 0.245 0.001 (0.011)Mother has less than vocational school diploma 0.323 0.326 -0.003 (0.012)Mother has a foreign nationality 0.063 0.069 -0.006 (0.006)Live in publicy subsidized housing 0.144 0.148 -0.004 (0.009)Own their home 0.603 0.617 -0.014 (0.012)Observations 3258 3164(in percent) 50.7% 49.3%


Most variables are binary, means thus indicate the share of students for whom the item is true.! p < 0.1, !! p < 0.05, !!! p < 0.01

Table 7 shows the mean for both genders as well as the t-test of di!erences for familyinputs prior to primary school or those that are invariant. Among a much longer list of avail-able background characteristics, the table shows those that are used in the decompositionsexplaining mathematics achievement. All variables are binary, allowing either the answer yes(numbered “1”) or no (numbered “0”), except the number of children in the family, wherethe answer corresponds to the number of brothers and sisters of each student. All variables inTable 7 were taken from the parent questionnaire when available. For students whose parentsdid not complete the questionnaire, information about the socio-professional category of the

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parents as well as the number of children in the family was imputed from the informationprovided by the school principal.

Table 7 shows that students have on average two or three brothers and sisters. Moreover,while more than 90% of fathers work, less than 70% of mothers do so. Approximately 35% offathers occupy an intermediate or higher socio-professional category. One fourth of mothershave obtained a university diploma, while one third have not completed high school. Almostseven percent of mothers have a foreign nationality. While about 60% of children come fromfamilies who own the home they live in, about 15% live in publicly subsidized housing. Weexpect to see no di!erences in background characteristics between boys and girls becausethere is little evidence that parents from a particular background prefer one gender overthe other3. Indeed, as the t-tests of di!erences in Table 7 show, none of the backgroundcharacteristics di!er between boys and girls.

Table 8: Summary statistics and t-test for maternal school inputs

Boys Girls t-test of di!erencesMore than 3 years maternal school attendance 0.231 0.245 -0.014 (0.011)Child didn’t like maternal school 0.085 0.042 0.043$$$ (0.006)Child didn’t learn a lot in maternal school 0.118 0.091 0.026$$$ (0.008)Observations 3258 3164(in percent) 50.7% 49.3%



A di!erence between boys and girls can be observed when it comes to subjective judgementsby parents on their child’s maternal school attendance. Table 8 shows the summary statisticsand t-test for the three variables concerning the length of maternal school attendance as wellas the opinion of parents. These variables enter the regression analysis performed in Section 6as “maternal school input”.

While 99% of all students go to maternal school before entering primary school, slightlyless than one fourth stays in maternal school for more than three years. This figure does notdi!er between boys and girls. However, parents of girls have a considerably more positiveimage about their child’s maternal school achievement than parents of boys. While only 4%of girls’ parents think that their child did not like maternal school, twice as many parents ofboys think so. Moreover, while 9% of girls’ parents think that their child did not learn a lotin maternal school, 12% of boys’ parents are convinced of this. Both di!erences are stronglystatistically significant.

Family-induced school inputs are what I call inputs which the child receives thanks to beingin school, but which are actually decided by his parents. These inputs comprise notably thecontact between parents and primary school as well as distance between home and school.Table 9 summarizes the means and t-tests for the four variables which will be used in theregression analysis in Section 6. The summary statistics were calculated for the subsampleof students for whom the gender gap at the entry into CE2 (third grade) was computed. Asattrition is relatively high and as values are missing for many students, the sample size forthe analysis of the gender gap in CE2 decreases to 3731 students.

There is no di!erence between boys and girls with respect to the probability that one ofthe parents accompanies a school excursion or picks up his child after school. According to

3Sohn (2010) discusses the unlikely possibility that mothers from a better socio-economic background aremore likely to have sons than daughters and cites Almond and Edlund (2007), who claims to have foundevidence for exactly that.

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Table 9: Summary statistics and t-test for family induced school inputs

Boys Girls t-test of di!erencesSchool more than 15min from home 0.092 0.076 0.017$ (0.009)Parent met with school psychologist 0.045 0.027 0.018$$$ (0.006)Parent accompanied school excursion 0.266 0.268 -0.002 (0.014)Parent picked up child after school 0.860 0.862 -0.002 (0.011)Observations 1866 1865(in percent) 50% 50%



the data, boys are slightly more likely to live in a home which is more than 15 minutes fromschool. While this di!erence is statistically significant at the 10% level, it is very small inmagnitude and possibly a pure coincidence. A more significant di!erence between gendersconcerns the probability that a parent met the school psychologists. While 4.5% of boys’parents met the school psychologist in the year prior to the administration of the parentquestionnaire (meaning in the first year of primary school), only 2.7% of girls’ parents did so.

Table 10 summarizes the means and di!erences of family inputs after the entry into primaryschool. These variables provide a good idea on the educational input from parents and explainan important part of the variation in achievement, as will be shown in Section 6. Of course, itis likely that the educational input was similar before and after the entry into primary school.However, as the parent questionnaire was administered only in the second school year, wecannot exclude the possibility that the input was influenced due to the experience made inthe first year of school. We therefore cannot ascertain that the family input was determinedprior to the first measure of school achievement. Hence, these variables cannot be includedin the model explaining primary school input.

There is no significant di!erence between the share of parents who regularly verify if theirchild studies well or discuss class activities with their child. Moreover, mothers are regularlyin charge of their child in as many boys’ families than girls’ families. Also, there is nodi!erence in the share of boys and girls who follow additional public or private coaching.Finally, boys and girls are just as likely to do manual activities and to play parlor games withtheir family.

For all other family input variables, there is a strongly significant di!erence between boysand girls. A strongly stereotypical gender pattern exists for both the parents’ opinion abouttheir child’s primary school achievement as well as the afternoon activities of children. Con-cerning their child’s school achievement, parents seem to be strongly convinced that girlsachieve better and provide a stronger e!ort than boys. Boys’ parents are not only about 5%more likely to check their child’s exercise book and school bag, they are also more than 7%more likely to be convinced that their child fails because he does not listen well in class. Onthe other hand, the parents of girls are more than 5% more likely to think that di"cultiesare due to the fact that school exercises are hard. Moreover, parents of girls are more than7% more likely to think that their child is good in writing, whereas parents of boys are morethan 11% more likely to think that their child is good in mathematics. This observation isinteresting as the opinion about di!erences in mathematics achievement is not at all justifiedgiven the results evaluated at the entry into first as well as third grade. This observationthus gives a strong hint towards the interpretation that gender di!erences in mathematicsachievement are at least partly due to stereotypical thinking at home.

Concerning free-time activities, gender di!erences are stereotypical as well. While boysare more likely to do sports with family members or to be enrolled in a sports club (here thedi!erence is 22%!), girls are more likely to read a story with family members or be enrolled

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Table 10: Summary statistics and t-test for family input after the entry into primary school

Boys Girls t-test of di!erencesMother regularly in charge of child 0.733 0.738 -0.005 (0.014)Parents regularly verify if child studies well 0.713 0.694 0.018 (0.015)Parents discuss class activities with child 0.730 0.725 0.004 (0.015)Parents check exercise book and schoolbag 0.790 0.743 0.047$$$ (0.014)Child follows additional public coaching 0.019 0.017 0.002 (0.004)Child follows additional private coaching 0.020 0.023 -0.003 (0.005)Di"culties because child doesn’t listen well in class 0.645 0.572 0.073$$$ (0.016)Child has di"culties because it’s hard 0.584 0.638 -0.053$$$ (0.016)Parents think child is good in writing 0.735 0.810 -0.075$$$ (0.014)Parents think child is good in mathematics 0.384 0.270 0.113$$$ (0.015)Child does sports or walking with family member 0.577 0.507 0.070$$$ (0.016)Child does manual activities with family 0.235 0.249 -0.014 (0.014)Child plays parlor games with family member 0.274 0.263 0.011 (0.015)Child reads a story with family member 0.524 0.572 -0.048$$$ (0.016)Child enrolled in sports club 0.716 0.497 0.219$$$ (0.016)Child enrolled in music school 0.128 0.387 -0.260$$$ (0.014)Child regularly watches TV before school 0.299 0.241 0.058$$$ (0.015)Observations 1866 1865(in percent) 50% 50%



in a music school. Moreover, boys are 6% more likely to watch TV in the morning beforeschool. Interestingly – and not represented here – no gender di!erence is reported on thetime spent playing computer games.

Table 11 shows the summary statistics of school inputs in the first two years of school,which were used to analyze mathematics achievement as measured at the entry into CE2(3rd grade). The first three variables indicate the number of students, the number of foreignstudents and the number of students repeating the class. Each of these variables is eitherobserved in 1st grade (CP) or 2nd grade (CE1). We can see that students were on averagein classes with 23 students. Moreover, less than one student per class is foreign. Also, lessthan one student in each class is repeating 1st grade.

The last four variables are binary and indicate additional school and class characteristics.Approximately 8% of the students are in a school which belongs to a priority education zone(Zone d’education prioritaire, ZEP), one of France’s school districts that receive particularattention and support because it accommodates families from a socially disadvantaged back-ground. Moreover, about 15% of all students are enrolled in a private primary school andmore than 31% live in a city with more than 100,000 people. While all of these variablesdo not di!er between boys and girls, the final variable, indicating the share of girls in classdoes. While girls are on average in classes with a girl share of 51%, the average girl share inthe classes of boys is 4% lower. The di!erence is statistically significant. Given that the girlshare for classes of boys is farther from 50% than the one of girls, we could think that boysare more likely to go to a non-mixed school.

Finally, Table 12 shows the same school input statistics as Table 11, this time for the lastthree years of primary school. These values will be used for the examination of mathematicsachievement at the entry into 6th grade, as these figures are more recent than those fromthe first two years of school. Note the further decrease of the sample size, which is due to

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Table 11: Summary statistics and t-test for school input in the first 2 years of primary school

Boys Girls t-test of di!erencesNumber of students in CE1 23.264 23.198 0.066 (0.121)Number of foreign students in CP 0.928 0.946 -0.018 (0.063)Number of repeating students in CP 0.930 0.953 -0.023 (0.039)School in priority education zone (ZEP) 0.084 0.070 0.013 (0.009)Child went to private school in CE1 0.150 0.141 0.009 (0.012)City with more than 100,000 people 0.313 0.324 -0.011 (0.015)Girl share in CE1 0.469 0.508 -0.039$$$ (0.003)Observations 1866 1865(in percent) 50% 50%



Table 12: Summary statistics and t-test for school input in the last 3 years of primary school

Boys Girls t-test of di!erencesNumber of students in CM2 24.476 24.332 0.144 (0.139)Number of foreign students in CM2 0.773 0.738 0.035 (0.063)School in priority education zone (ZEP) 0.055 0.044 0.011 (0.008)Child went to private school in CM2 0.169 0.147 0.021 (0.014)Child lives in rural area 0.192 0.199 -0.007 (0.015)City with more than 100,000 people 0.295 0.298 -0.004 (0.017)Girl share in CE2 0.468 0.512 -0.045$$$ (0.004)Observations 1446 1458(in percent) 49.8% 50.2%



further attrition from the sample between 3rd and 6th grade. Note also that the variableindicating the number of repeating students in the class was replaced by a binary variablestating whether the student lives in a rural area.

The pattern of the school inputs in the last three years of primary school is very similar tothe one observed in the first two years. The number of students slightly increases to almost 25students per class, while the number of foreign students decreases. A lower share of studentsgoes to a school in a priority education zone, which might be due to the fact that attritionfrom the sample primarily concerns students from less favorable social backgrounds. Table 12also shows that about 20% of students live in rural areas, which is consistent with France’surban population share of about 80%. The share of students enrolled in a private school, theshare of students living in a city with more than 100,000 people as well as the girl share aresimilar to those observed for the first two years of primary school (Table 11). Again, boysare enrolled in classes with a girl share of 47% while the girl share in girls’ classes is 51%.

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5 The gender gap

results obtained with a ttest, Stata command: ttest var, by(gender)results obtained with a ttest, Stata command: ttest var, by(gender)results obtained with a ttest, Stata command: ttest var, by(gender)results obtained with a ttest, Stata command: ttest var, by(gender)

maths av french av

diff cp 0,014 0,199

sd 0,0206 0,0211

diff ce2 -0,022 0,275

sd 0,023 0,0228

diff 6e -0,242 0,321

sd 0,0257 0,024

CP CE2 6e

upper bound 0,054376 0,02308 -0,191628

Maths mean 0,014 -0,022 -0,242

lower bound -0,026376 -0,06708 -0,292372

0 0 0

upper bound 0,240356 0,319688 0,36804

French mean 0,199 0,275 0,321

lower bound 0,157644 0,230312 0,27396

geometry 0,019 0,051 -0,063

writing 0,152

verbal knowledge 0,223

reading 0,151

comprehension 0,053

numeration 0,004

measuring 0,006

numerical exercises -0,061

problem solving -0,053

language tools 0,361

comprehension ce2 0,122

comprehension 6e 0,248

writing ce2 0,26

writing 6e 0,38

writing numbers -0,297

treating operators -0,192

solving numerical problemssolving numerical problems -0,084

treating informationtreating information -0,348

analyzing a situationanalyzing a situation -0,276

applying a techniqueapplying a technique 0,044

producing an answerproducing an answer -0,076

researching informationresearching information -0,192

using knowledge -0,256

knowing to write 0,365

knowing to read 0,241

language tools 6e 0,226

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Figure 3: Mean gender gap in mathematics (light grey) and French (dark grey) – Di!erencebetween the score of girls and boys

This section presents gender di!erences at the standardized evaluations which were con-ducted at the entry into “classe preparatoire” (1st grade), at the entry into CE2 (3rd grade)and at the entry into “sixieme” (6th grade). The aim is to provide an overview of the dif-ferences between boys and girls at the mean and for the entire distribution. For now, Ido not attempt to examine the reasons for potential di!erences, which will be done in thefollowing section. Before pointing out the detailed gender di!erences for each sub-score, Fig-ure 3 shows the mean gender gap for all mathematics and French scores. Gender di!erencesin mathematics scores are indicated in light grey, while they are depicted in dark grey forFrench scores. The graph indicates the mean di!erence between girls and boys in standarddeviations, a positive value thus indicates a higher score for girls. Lines show the gender gapfor the average French and mathematics result, with a 95%-confidence interval, whereas thesmall circles indicate gender di!erences for the various sub-scores in these evaluations.

5.1 The gender gap at the entry into 1st grade

At the entry into primary school, cognitive skills were evaluated on 10 tasks, which can begrouped into three categories: General tasks, French and mathematics. Table 13 summarizesgender di!erences in the performance on these cognitive skills. For each of the evaluations, thecolumn “Performance of girls compared to boys” indicates whether girls score higher or lowerthan boys. A value of “0” stands for an absence of gender di!erences. “++” indicates thatgirls score more than a tenth of a standard deviation higher than boys, while “+” states thatgirls scare only slightly better. Table 13 provides no details for the gender gap throughoutthe distribution because di!erences between boys and girls are the same at each percentileexcept in the cases indicated in the last column. We can make four observations.

First, there are no gender di!erences in mathematics. Figure 4 shows the di!erence betweenthe score of boys and girls for each percentile of the distribution in the evaluation of numbersand geometry. The di!erence is expressed in terms of standard deviations. As one can see,girls and boys obtain the same scores throughout the entire distribution. The same is truefor the second evaluation of mathematics at the entry into primary school, which consists ofexercises involving counting and simple calculations.

Second, girls score better on all items evaluating language skills except for oral comprehen-sion. The gender gap is particularly strong in the test of verbal knowledge, where it exceeds

36

Table 13: Overview of gender di!erences for cognitive skills at the entry into 1st grade

Category TaskPerformance ofgirls comparedto boys

Remark

General Spatial concepts 0Girls score strongly bet-ter at lowest %-tiles

General knowledge 0 -Temporal concepts 0 -Technical knowledge – – Strong di!erence

French Writing ++ -Verbal knowledge ++ Strong di!erence

Reading ++ -

Oral comprehension 0Girls score strongly bet-ter at lowest %-tiles

Mathematics Numeration 0 -Numbers/geometry 0 -

++: Girls score more than 0.1 standard deviations above boys

+: Girls score less than 0.1 standard deviations above boys

0: No gender di!erences

– –: Girls score more than 0.2 standard deviations below boys

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Figure 4: Gender Gap in 1st grade: mathematics (numbers and geometric figures)

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Figure 5: Gender Gap in 1st grade: verbal knowledge (left), oral comprehension(right)

0.2 standard deviations. The left graph of Figure 5 shows that girls score approximately0.22 standard deviations higher than boys, a di!erence which is significant at the 1%-level.The graph also shows that there is no particular pattern of gender di!erences throughoutthe distribution. The peaks at some percentiles are due to the fact that there is a limitednumber of possible points. Girls score slightly more than 0.15 standard deviations above boysin writing and reading. In the evaluation of oral comprehension, girls score strongly betterat the lowest percentiles. This can be seen in the right graph of Figure 5. While there isno gender di!erence for any student above the 20th percentile, the weakest girls score abouthalf a standard deviation above boys.

Third, the only gender di!erence among the four evaluations of general skills can be ob-served for technical knowledge. In the latter, girls score on average 0.26 standard deviationsbelow boys, which is the largest gender gap of the entire test. This strong di!erence is stablethroughout the distribution. Similar to what can be observed for some of the family inputcharacteristics described in Section 4, the strong gender gap in the favor of boys for technicalskills is consistent with the existence of stereotypical behavior.

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Figure 6: Gender Gap in 1st grade: knowledge of spatial concepts

Moreover, in the evaluation of spatial competences, girls in the lowest percentiles of thedistribution score considerably higher than boys. Figure 6 shows the score di!erence between

38

boys and girls for spatial skills for the entire distribution. We can see that girls score almosthalf a standard deviation above boys at the lowest 5 percentiles of the distribution, whereasno gap exists for any quantile above that. This finding contradicts the claim of many authors(Niederle and Vesterlund 2010), according to which boys develop better spatial skills thangirls. However, more than 42% of the students scored the full score of 100 points. If a gendergap does not exist for the largest part of the distribution, this might also be due to the factthat di!erences between the most skilled 40% of the students are not measured.

Finally, all of the described gender di!erences are entirely stable throughout the distribu-tion of scores except for spatial competences and oral comprehension, as stated in the lastcolumn of Table 13. In the latter, a gender gap favoring girls exists for the lowest percentiles,while there is no gap for the rest of the distribution.

5.2 At the entry into 3rd grade

Table 14: Gender di!erences in mathematics and French at the entry into CE2 (3rd grade)

Task Performance of girls compared to boysMean Lowest 10 percentiles Median Highest 10 percentiles

Mathematics (evaluation 1999, 94% of students)Average score 0 (+) (–) –Geometry + 0 0 0Measuring 0 0 0 0

Numerical exercises – 0 (– –) – –Problem solving – 0 0 – – –

French (evaluation 1999, 94% of students)Average Score ++ +++ ++ +Comprehension + ++ + (+)Language tools +++ +++ +++ ++

Writing ++ +++ ++ 0

(Parentheses indicate that di!erences are not statistically significant)

+++: Extreme gender gap: Girls score more than 0.30 std. deviations above boys

++: Large gender gap: Girls score 0.15 - 0.30 std. deviations above boys

+: Small gender gap: Girls score 0.05 - 0.15 std. deviations above boys

0: Gender di!erences of less than 0.05 standard deviations

–: Small gender gap: Girls score 0.05 - 0.15 std. deviations below boys

– –: Large gender gap: Girls score 0.15 - 0.3 std. deviations below boys

– – –: Extreme gender gap: Girls score more than 0.3 std. deviations below boys

Table 14 summarizes the gender gap in mathematics and French for all sub-scores of theevaluation at the entry into CE2 (3rd grade). As described in Section 4, the standardizedevaluation at the entry into 3rd grade was subdivided into four areas in mathematics as wellas into three areas in French. As indicated in the head of both sections of the table, theinformation concerns those students who took the 3rd-grade evaluation in 1999, which meansat the regular date. 6% of the students took the test one year later because they were obligedto repeat either the 1st or the 2nd grade. In addition to the mean gender di!erence, Table 14shows the gender gap for the median as well as the lowest and highest 10 percentiles of thedistribution. Just like in Table 13, summarizing the scores at the entry into 1st grade, thenumber of pluses or minuses indicates by how much girls score higher or lower than boys,

39

as explained in the legend below the table. Statistically insignificant gender di!erences areindicated in parentheses.

Table 14 leads us to the following four observations. First, there is a small gender gapto the advantage of boys on the average mathematics score, which is mainly due to genderdi!erences for numerical exercises and problem solving. Boys and girls score exactly the samein measuring at each part of the distribution.

The score in geometry almost does not vary between genders. On average, girls score 0.05standard deviations above boys. The distributional graph (left graph in Figure 7) showsthat there is no particular pattern or a systematic gap at some quantiles of the distribution.As some authors (Niederle and Vesterlund 2010) attribute the mathematics gender gap tohigher spatial competences of boys, we might have expected that the gap in 3rd grade canin particular be explained with di!erences in the geometry score. However, as Table 14 andFigure 7 show, this does not seem to be the case in the French Primary School Panel.

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Figure 7: Mathematics gender gap in 3rd grade: geometry (left), numerical skills (right)

When it comes to numerical skills, the gender gap is considerably more pronounced. Theright graph in Figure 7 shows the gender di!erences for this test throughout the entiredistribution. While girls in the lower half of the distribution score exactly as well as boys, theyscore close to 0.2 standard deviations below boys at the higher end of the distribution. Thedi!erence is statistically significant. The scores on problem solving follow a similar patternas those on numerical skills, notably with a strong disadvantage for girls at the higher endof the distribution. While girls at the highest 10 to 20 percentiles score approximately half astandard deviation below boys, whereas there is no di!erence at most other quantiles.

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Figure 8: Gender Gap in French 3rd grade: Language tools (left), Average score (right)

40

Table 15: Mean and standard deviation for scores at the evaluation in 3rd grade

Boys GirlsMathematics Average score 0.000 (1.000) -0.022 (0.963)

Geometry -0.000 (1.000) 0.051 (0.979)Measuring -0.000 (1.000) 0.006 (0.995)Numerical Exercises 0.000 (1.000) -0.061 (0.943)Problem solving -0.000 (1.000) -0.053 (0.948)

French Average score -0.000 (1.000) 0.275 (0.942)Comprehension -0.000 (1.000) 0.122 (0.966)Language tools -0.000 (1.000) 0.361 (0.941)Writing 0.000 (1.000) 0.260 (0.902)Observations 7321

Columns: mean and standard deviation for boys and girls.

Second, the gender gap is much stronger in French, where girls considerably outperformboys. The advantage of girls over boys in the French test is in particular due to strongdi!erences in the score of language tools. The left graph of Figure 8 shows the di!erencesacross the distribution for the exercises on language tools. We can see that the gender gap isparticularly strong at the lower end of the distribution, where it can reach up to 0.6 standarddeviations to the advantage of girls. To a lesser extent, girls also considerably outperformboys in comprehension and writing.

Third, the gender gap is not constant throughout the distribution of scores. In bothmathematics and French, girls are particularly strong compared to boys in lower quantiles.In the numerical exercises and problem solving evaluation of the mathematics test, girls arestrongly weaker at the top of the distribution, while their scores do not di!er from boys orare even better at the low end of the distribution. In all French sub-scores, girls extremelyoutperform boys in low percentiles, while in higher quantiles the di!erence only remainssignificant for the language tools examination. This can be seen in a striking way in theright graph of Figure 8, which shows the gender di!erences for the average French score. Atthe lower end of the distribution, girls obtain an average French mark which is 0.4 standarddeviations above boys, whereas the di!erence reduces to below 0.2 standard deviations at theupper end of the distribution.

Finally, due to the relative strength of girls compared to boys in lower quantiles, the scoresof girls have a lower variance. This is true in all mathematics and French test scores. Table 15shows the mean and standard deviation of all scores of the 3rd-grade evaluation. Note thatall scores were normalized such that boys score with a mean of 0 and a standard deviation of1. In addition to the gender di!erence in terms of standard deviation, Table 15 makes clearthat the variance of girls’ scores is inferior to the one of boys for all evaluations.

5.3 At the entry into 6th grade

The evaluation at the entry into 6th grade consists of twice as many sub-categories as theevaluation at the entry into 3rd grade. In mathematics, each exercise of the evaluation canbe attributed to one of the following fields: geometry, writing numbers, treating operators,solving numerical problems and treating information. At the same time, each exercise assessesone of the five following skills: Analyzing a situation, Applying a technique, Producing ananswer, Researching information and Using knowledge. Together with the overall averagescore, this provides us with 11 scores assessing mathematics achievement at the entry into6th grade. The French evaluation consists of five parts: Comprehension, Spelling words,Knowing to read, Language tools and Writing a text. Table 16 gives an overview of thegender gap and its distribution for each of these sub-scores.

41

Table 16: Gender di!erences in mathematics and French at the entry into 6th grade

Task Performance of girls compared to boysMean Lowest 10 percentiles Median Highest 10 percentiles

Mathematics (evaluation 2002, 79% of students)Average score – – – – – – –Geometry – – – 0

Writing numbers – – – – – – –Treating operators – – – – – – –Numerical problems – 0 – – 0Treating information – – – – – – – – – – –Analyzing a situation – – – – – – – – –Applying a technique 0 0 0 0Producing an answer – 0 – 0Research information – – – – – – – –Using knowledge – – – – – – –

French (evaluation 2002, 79% of students)Average Score +++ +++ +++ ++Comprehension ++ +++ ++ 0Spelling words +++ +++ +++ ++

Reading ++ +++ ++ +Language tools + +++ ++ +Writing a text +++ +++ +++ +

+++: Extreme gender gap: Girls score more than 0.30 std. deviations above boys

++: Large gender gap: Girls score 0.15 - 0.30 std. deviations above boys

+: Small gender gap: Girls score 0.05 - 0.15 std. deviations above boys

0: Gender di!erences less than 0.05 standard deviations

–: Small gender gap: Girls score 0.05 - 0.15 std. deviations below boys

– –: Large gender gap: Girls score 0.15 - 0.3 std. deviations below boys

– – –: Extreme gender gap: Girls score more than 0.3 std. deviations below boys

In summary, we can make the following four observations. First, while the gender gap inFrench has about the same magnitude as in 3rd grade, it strongly increases in mathematics.Table 17 presents the mean normalized test scores and their standard deviation at the entryinto 6th grade for both boys and girls. The statistics for all sub-scores in mathematicsand French are indicated. Again, test scores were normalized to have a mean of 0 anda standard deviation of 1 for boys. This allows to easily detect di!erences in mean andstandard deviation for girls. We can observe that the average score in French is on average0.32 standard deviations higher for girls than for boys, a value similar to the 0.28 standarddeviations in 3rd grade (see Table 15). Contrarily, in mathematics, where only a small gendergap existed on the average mathematics score in 3rd grade, girls score almost a quarter of astandard deviation below boys at the end of primary school.Second, both in French and in mathematics, the gender gap decreases with percentiles.

This is di!erent to what we observed in CE2 (3rd grade), where girls scored particularly wellat the lower end of the distribution. It can be seen in Figure 9, which shows the gender gap forthe entire distribution of average mathematics (left graph) and French (right graph) scores.The gender gap is particularly strong at the lower end of the distribution. In French, this

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Table 17: Mean and variance for scores at the evaluation in 6th grade

Boys GirlsMathematics Average score 0.000 (1.000) -0.242 (1.040)

Geometry 0.000 (1.000) -0.063 (1.017)Writing numbers -0.000 (1.000) -0.297 (1.045)Treating operators -0.000 (1.000) -0.192 (0.981)Solving numerical problems 0.000 (1.000) -0.084 (1.067)Treating information 0.000 (1.000) -0.348 (1.053)Analyzing a situation 0.000 (1.000) -0.276 (1.025)Applying a technique -0.000 (1.000) 0.044 (0.974)Producing an answer 0.000 (1.000) -0.076 (1.015)Researching information -0.000 (1.000) -0.192 (1.049)Using knowledge 0.000 (1.000) -0.256 (1.034)

French Average score 0.000 (1.000) 0.321 (0.926)Comprehension 0.000 (1.000) 0.248 (0.917)Knowing to write -0.000 (1.000) 0.365 (0.926)Knowing to read -0.000 (1.000) 0.241 (0.935)Language tools 0.000 (1.000) 0.226 (0.945)Writing a text -0.000 (1.000) 0.380 (0.913)

Observations 6401Columns: mean and standard deviation for boys and girls.

All variables are normalized on the score of boys.

leads to the same pattern which we already observed in CE2 (see the right graph in Figure 8).However, in mathematics, the gender gap throughout the distribution is reversed. On the leftgraph of Figure 9, one can see that girls score up to almost 0.4 standard deviations below boysaround the 20th percentile, while the gap reaches only about 0.1 standard deviations amongthe highest 20 percentiles. According to Ellison and Swanson (2010) and other authors, thegender gap is particularly strong at the higher end of the distribution. Interestingly, the datafrom the French Primary School Panel suggests the inverse.

The reversal of the distributional pattern of the gender gap in mathematics between 3rdand 6th grade might be the consequence of a di!erential evolution of the gap. At the top ofthe distribution, girls score between 0.05 and 0.1 standard deviations lower than boys alreadyin 3rd grade, a gap which does not considerably increase until 6th grade. However, at thelower end of the distribution, no gap exists in 3rd grade, whereas girls score strongly weakerin 6th grade. This could mean that the gender gap develops earlier among better students.

Given that girls score higher in French and lower in mathematics than boys and giventhat the gender gap decreases strongly throughout the distribution, we expect the varianceof girls’ scores to be higher than those of boys in mathematics and lower in French. Indeed,Table 17 confirms this intuition. The scores being normalized to a variance of 1 for boys,we can see that it is higher for girls in almost all mathematics evaluations and considerablylower in French.

Table 16 shows to what extent this distributional pattern can be found among the othersub-scores. One can see that the pattern is similar for all French scores. In mathematicshowever, the gender gap only decreases with quantiles for some sub-scores. The pattern isdi!erent for numerical problems, analyzing a situation, applying a technique and producingan answer. Here, the gender gap is weaker at both extremes and strongest at the median.Note that no gender gap can be observed at all for the sub-score evaluating the applicationof a technique.

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Figure 9: Gender Gap for the entire distribution at the entry into 6th grade: Average scoremathematics (left), Average score French (right)

Third, the source of the gender gap in French is opposite to what was observed in CE2.While in CE2 the gap was high in the evaluation of the knowledge of language tools and lowfor comprehension, the inverse is true in 6th grade. As Table 17 shows, the lowest gender gapnow concerns the scores of language tools, where girls only score 0.23 standard deviationsabove boys. The gap is highest for writing a text. Still, di!erences between sub-scores arenot extreme and girls undoubtedly score considerably higher than boys in all evaluations ofFrench.

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Figure 10: Gender Gap in 6th grade: Mathematics (Geometry)

Finally, in mathematics and similar to 3rd grade, the gender gap is relatively low in geome-try. On average, girls score only 0.06 standard deviations below boys (see Table 17). Figure 10shows the gender gap for the entire distribution. We can see that the gap is generally smalland even inexistent at some quantiles. In particular, no gender gap exists for students amongthe highest 20 percentiles. Again, this contradicts the widely cited theory according to whichthe advantage of boys in mathematics results from better spatial skills, which also claimsthat the gender gap is particularly strong in geometry (Niederle and Vesterlund 2010).

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6 Results of the decomposition

This section presents the results from the detailed Oaxaca-Blinder decomposition describedin Section 3 for mathematics achievement at the entry into 1st, 3rd and 6th grade. For allthree evaluations, I will present the regression estimates according to the model described inSection 3 before presenting and interpreting the detailed decomposition.

6.1 Evaluation at the entry into primary school (CP)

As described above, I suppose that achievement at the entry into primary school depends onfamily inputs prior to school as well as on inputs resulting from maternal school. Please referto Section 4 for a detailed description of the data. Family inputs can be attributed to fivecategories: family composition, parents’ job, parents’ education, parents’ origin and livingconditions. Two types of variables describe maternal school inputs: duration of maternalschool attendance and parents’ opinion about maternal school. The French Primary SchoolPanel contains numerous variables for each of these categories. However, I restrict the theregression to variables which are significant and representative. Moreover, variables werechosen to maximize the number of individuals for whom each information is available.

As discussed in Section 3, in addition to family and maternal school inputs, I use thescores obtained in the evaluation of non-cognitive skills, which was conducted by teachers, inorder to explain mathematics performance at the entry into primary school. In accordancewith the model described in Section 3, we assume that once these variables are included asexplanatory variables, the residual individual-specific e!ect is uncorrelated with the residualsof the achievement production function.

Table 18 shows the linear regression models explaining the average score in mathematicsat the entry into primary school with the above-mentioned family and maternal school in-puts as well as non-cognitive skills separately for boys and girls. Here, I use the averagemathematics score as the dependent variable, meaning the average of the two mathematics-related evaluations which were conducted at the entry into 1st grade (simple calculationsas well as numbers and geometry). Please refer to Section 4 for a detailed description ofthe competences evaluated in these tests. Like all achievement results in this paper, scoreswere normalized such that boys score a mean of 0 and a standard deviation of 1. Thus, allcoe"cients indicate marginal changes in terms of standard deviations. For better readability,the three categories of explanatory variables are separated by horizontal lines. The variablesincluded in Table 18 explain more than 33% of the variation of mathematics achievement forboys and more than 38% for girls.

The coe"cients relating family and maternal school inputs as well as non-cognitive skillsto mathematics achievement at the entry into primary school show a typical pattern. Allvariables indicating a more advantaged socio-economic background have a positive impacton the outcome. Considerable di!erences exist between genders. While the mathematicsachievement of boys is strongly determined by the professional situation of their parents andto a lesser extent by the parents’ education, girls see their achievement mainly influenced bytheir mother’s education and the family’s living conditions.

An interesting observation is that parental employment has a positive e!ect on mathemat-ics achievement of boys. The average score in mathematics at the entry into primary schoolincreases by almost 0.09 standard deviations if the mother works and by 0.13 standard devi-ations if the father works. Such a strongly positive e!ect is not unanimously found in priorresearch. The relation between parental employment and child achievement is subject todebate among researchers on its own. Most studies find, if anything, a negative impact inparticular of maternal employment on intellectual ability. Ruhm (2004) summarizes exist-ing research on the e!ect of parental employment on the cognitive achievement of children.He finds research advocating both positive and negative e!ects. Ruhm (2004) concludes by

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Table 18: Mathematics achievement at the entry into 1st grade (CP)

Average score mathematics in CPBoys Girls

Number of children in family 0.002 (0.013) -0.021 (0.014)Mother works 0.088*** (0.033) 0.049 (0.033)Father works 0.130** (0.064) 0.058 (0.064)Socio-professional category 3 or 4 (1997, father) 0.127*** (0.030) 0.024 (0.033)Mother has obtained university diploma 0.062* (0.035) 0.073** (0.037)Mother has less than vocational school diploma -0.107*** (0.036) -0.112*** (0.036)Mother has a foreign nationality -0.003 (0.071) -0.071 (0.063)Live in publicy subsidized housing -0.017 (0.049) -0.121** (0.048)Own their home 0.036 (0.034) 0.038 (0.036)More than 3 years maternal school attendance 0.091*** (0.032) 0.077** (0.032)Child didn’t like maternal school -0.154*** (0.056) -0.039 (0.080)Child didn’t learn a lot in maternal school -0.290*** (0.048) -0.238*** (0.058)Self-confident during school activities 0.117*** (0.041) 0.182*** (0.043)Never fails due to excessive confidence 0.004 (0.030) -0.075** (0.031)Capable of regular attention 0.102** (0.043) 0.069 (0.048)Actively participates in group activities 0.033 (0.042) 0.080* (0.046)E"ciently completes tasks 0.205*** (0.049) 0.133** (0.054)Autonomous 0.029 (0.047) 0.126** (0.054)Never fatigues during school activities 0.053* (0.029) 0.037 (0.030)Consciously intervenes in class discussions 0.157*** (0.042) 0.102** (0.044)Anticipates and is organized 0.157*** (0.046) 0.037 (0.047)Good linguistic level compared to class average 0.266*** (0.039) 0.227*** (0.041)Requires support and encouragement -0.400*** (0.044) -0.389*** (0.044)Requires corrective measures and warning -0.261*** (0.046) -0.408*** (0.064)Constant -0.742*** (0.104) -0.489*** (0.104)R-squared 33.4% 38.4%Observations 3258 3164

Standard errors in parentheses

* p < .1, ** p < .05, *** p < .01

summarizing that maternal employment has a negative e!ect, which may be o!set when thechild becomes 3 to 4 years old. Desai, Chase-Lansdale and Michael (1989) find a negativeimpact of maternal employment on the cognitive achievement of boys from advantaged socio-economic backgrounds. Similarly, Moore and Driscoll (1997) state that a positive impact ofmaternal employment can be observed only among mothers whose wage is high enough. Onthe contrary, a small and rather positive e!ect in particular among girls has been detectedby Goldberg, Prause, Lucas-Thompson and Himsel (2008).

Unsurprisingly, maternal school input has a strong e!ect on achievement at the entry intoprimary school. This is true both for the duration of maternal school as well as the opinionof parents about their child’s experience in this school. Boys’ achievement seems to be morestrongly influenced by maternal school than that of girls. The coe"cients of all three maternalschool input variables are considerably higher in absolute value for boys. For example, if theparents of boys think that their child did not learn a lot in maternal school, achievementis almost 0.3 standard deviations lower than that of boys whose parents do not think so,contrary to a di!erence of less than 0.25 standard deviations for girls.

The variables measuring non-cognitive skills, as evaluated by teachers at the entry intoprimary school, aim to account for the individual-specific e!ect or individual ability of each

46

student. They appear in the third section of Table 18. The pattern of coe"cients for non-cognitive skills does not reveal surprises. Variables indicating a better behavior or a higherlevel of skills such as motivation, participation or concentration are correlated with higherachievement. An interesting feature can be observed concerning self-confidence. While self-confidence is generally associated with an improvement of achievement between 0.1 and 0.2standard deviations for both boys and girls, the latter see their achievement diminish if theyare considered to “never fail due to excessive confidence”. The most important influence onmathematics achievement for both girls and boys stems from the last three non-cognitive skillmeasures of Table 18, indicating whether the child has a good linguistic level and whetherit requires support and encouragement as well as corrective measures. The mathematicsachievement of both girls and boys reduces by 0.4 standard deviations if the teacher considerssupport and encouragement necessary for the child. This confirms the importance of self-confidence, in addition to the coe"cients of the other measures of self-confidence (“self-confident during school activities” and “Never fails due to excessive confidence”).

Table 19: Decomposition of gender gap at the entry into 1st grade (CP)

(1) (2) (3)Average mark Simple calulations Numbers and geometry

Prediction for boys -0.000 (0.018) -0.000 (0.018) 0.000 (0.018)Prediction for girls 0.015 (0.017) 0.020 (0.017) 0.007 (0.017)Di!erence -0.015 (0.025) -0.020 (0.025) -0.007 (0.025)

Explained gender di!erencesFamily input -0.001 (0.004) -0.001 (0.003) -0.001 (0.003)Maternal school input -0.016$$$ (0.004) -0.014$$$ (0.003) -0.013$$$ (0.004)Noncognitive skills -0.095$$$ (0.014) -0.095$$$ (0.014) -0.070$$$ (0.012)Total -0.112$$$ (0.016) -0.109$$$ (0.016) -0.085$$$ (0.014)

Unexplained gender di!erencesFamily input 0.210$$ (0.107) 0.098 (0.109) 0.269$$ (0.117)Maternal school input -0.006 (0.013) 0.002 (0.014) -0.013 (0.015)Noncognitive skills 0.145$ (0.084) 0.109 (0.086) 0.144 (0.092)Intercept -0.253$ (0.136) -0.119 (0.139) -0.323$$ (0.149)Total 0.096$$$ (0.021) 0.090$$$ (0.021) 0.078$$$ (0.023)

Standard errors in parentheses! p < 0.1, !! p < 0.05, !!! p < 0.01

Table 19 shows the results for the decomposition of the mathematics gender gap at the entryinto 1st grade. The table shows the decomposition for three models: model (1) decomposesthe average mathematics achievement, while models (2) and (3) indicate the results for thetwo sub-scores (simple calculations as well as numbers and geometry). The upper part ofthe table shows the predicted values of the respective score for boys and girls as well asthe di!erence between them. A negative di!erence means that girls score better than boysbecause the di!erence is calculated as the average score of girls substracted from the oneof boys. Boys are predicted to score 0 in all three models, given that their achievementscores were normalized to 0. As indicated in the third row of Table 19, there is no significantdi!erence between the scores of boys and girls, as was stated in the description of the gendergap in Section 5.

In the second part of Table 19, we can see the explained part of the gender di!erence forthe three categories of explanatory variables described above. The explained part illustratesthe gender gap which is due to di!erences in explanatory variables between boys and girls.

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Coe"cients in this part of the table indicate how girls’ would score on average comparedto boys if they had the same achievement production function as boys. With other words,the coe"cients indicate the di!erence between boys and girls that would exist if their edu-cational achievement production function was the same, which would mean that no genderdiscrimination existed.

On aggregate, the fact that girls and boys have di!erent levels of explanatory variables –which is sometimes called “di!erent endowments” – leads to a gender gap of 0.11 standarddeviations on the average mathematics score to the advantage of girls. That means that ifthere was no unexplainable di!erence between the educational production function of boysand girls, girls would score more than 0.1 standard deviations better than boys in mathe-matics at the entry into primary school. As there are no gender di!erences in backgroundcharacteristics, this di!erence is entirely due to di!erences in non-cognitive skills as well asdi!erences in parental opinion on maternal school. We can see that with an identical ed-ucational outcome function, girls would score 0.095 standard deviations higher than boysbecause they are better rated in non-cognitive skills assessments. In addition, their mathe-matics achievement is predicted to be another 0.016 standard deviations above that of boysbecause parents of girls have a better opinion on their child’s time in maternal school.

In spite of the fact that, according to their level of non-cognitive skills, girls are predicted toscore better than boys, there is no gender gap in mathematics at the entry into primary school.This is due to the unexplainable part of the gender di!erence, which is represented in the thirdpart of Table 19. The unexplainable, or “structural”, part of the gender di!erence results fromdi!erences in the extent to which explanatory variables influence educational achievement.While the production function transforming maternal school input into achievement is thesame for boys and girls (the coe"cient for the unexplained di!erence is 0 for all three scores),family input has a much higher return for boys than for girls. On average, boys score 0.21standard deviations higher than girls with the same family input. This result is significantat the 5%-level. Moreover, for the same level of non-cognitive skills, boys obtain a score thatis almost 0.15 standard deviations higher than that of girls, a di!erence which is significantat the 10%-level.

Overall, the higher returns to family inputs and non-cognitive skills more than compensatethe predicted male disadvantage resulting from endowment di!erences. However, there is nogender gap in total because the intercept is a quarter of a standard deviation higher for girlsthan for boys. As stated in Section 3, a di!erence of the intercept can be interpreted as cap-turing di!erences in unobservable characteristics and their impact on achievement. Moreover,a di!erence of the intercept can be due to entirely unexplainable and thus purely discrimi-natory gender di!erences. In conclusion, we can therefore state that girls outperform boysdue to endowment in observable characteristics, as well as unobservable characteristics andentirely unexplainable discrimination, whereas boys outperform girls due to higher returnsto family input and non-cognitive skills.

An examination of the two sub-scores of mathematics at the entry into 1st grade does notyield interesting conclusions. The pattern of explained gender di!erences is similar for boththe score on simple calculations as well as the score on numbers and geometry. However,in the evaluation of numbers and geometry, boys have a particularly high return to familyinput, which indicates a predicted advantage of 0.27 standard deviations over girls. It iscompensated by an advantage in the intercept for girls that is even higher.

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6.2 Evaluation at the entry into 3rd grade (CE2)

At the entry into CE2, which is the third year of primary school, we still observe no gendergap in mathematics. As described in Section 5, girls score slightly better in geometry whileboys have an advantage in problem solving. Here, we examine to what extent these di!erencesas well as the absence of gender di!erences in the other mathematics scores can be attributedto di!erences in inputs. According to the model presented in Section 3, achievement at theentry into 3rd grade can be explained with a cumulative production function. In additionto the explanatory variables included in the explanation of achievement at the entry into 1stgrade, four new categories of variables influence achievement now: Family school contact,family input during primary school, primary school input and achievement at the entry intoschool. Among a large number of potentially important explanatory variables, I chose thosethat contribute most to the outcome as well as those which present strong di!erences betweenboys and girls.

Table 20 shows the OLS regression explaining the average mathematics score at the entryinto 3rd grade with the explanatory variables mentioned above. For better readability, somevariables were removed from the table. Horizontal lines separate the seven categories ofexplanatory variables: Family input prior to primary school, maternal school input, non-cognitive skills, parent induced school input in primary school, family input in primary school,school input in the first 2 years of primary school as well as achievement at the entry intoprimary school. The variables included in the regression explain more than 50% of thevariation of average mathematics achievement.

Table 20 allows us to make a number of observations regarding the link between the variousinputs and average mathematics achievement in 3rd grade. First of all, most of the coe"-cients are not surprising. For example, all indicators for an advantaged family backgroundhave a positive impact on mathematics achievement. Contrary to the evaluation at the entryinto 1st grade, parental employment has no e!ect on mathematics achievement in 3rd grade.The other background characteristics, such as the profession and education of parents as wellas living conditions influence achievement in a similar way as two years earlier. Interestingly,and contrarily to the previous evaluation, the fact that a child did not like maternal schoolseems be positively correlated with mathematics achievement. Boys who did not like mater-nal school score 0.14 standard deviations higher than boys who did. For girls, the di!erencereaches almost 0.2 standard deviations. Non-cognitive skills influence achievement in a sim-ilar, but less extreme way than in 1st grade. In particular, self-confidence seems to play alesser role.

Many of the variables describing the parents’ implication in their child’s school activitiesare correlated with mathematical achievement. For example, the fact that the child followsadditional coaching is a good predictor for di"culties in mathematics, especially for boys.Boys who follow either public or private coaching score more than 0.35 standard deviationsbelow boys who do not. Moreover, parents seem to have a more or less accurate opinionabout their child’s mathematics achievement, although in this case, cause and consequence isnot clear. Boys whose parents think their child is good in mathematics, score on average 0.34standard deviations higher than boys whose parents do not think so. For girls, the di!erenceis 0.29 standard deviations. However, we do not know whether parents have a positive opinionbecause their child is good, or whether their child is good because they motivate her or himby having a positive opinion. Parents’ actual attendance at their child’s school does not havea clear impact. Girls whose parents have accompanied school excursions score lower thangirls whose parents did not. Boys seem to score an eighth of a standard deviation lower iftheir parents pick them up after school.

A similarly unclear impact can be observed for family input regarding activities before orafter school. In general, home inputs have a stronger e!ect for girls than for boys. Surpris-ingly, girls and boys who do manual activities with their family as well as girls who readstories with a family member score less than those who do not.

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Table 20: Mathematics achievement at the entry into 3rd grade (CE2)

Average score mathematics in CE2Boys Girls

Number of children in family -0.031* (0.017) -0.021 (0.018)Mother works 0.038 (0.042) -0.021 (0.040)Father works -0.000 (0.093) -0.121 (0.082)Socio-professional category 3 or 4 (1997, father) 0.127*** (0.037) 0.086** (0.036)Mother has obtained university diploma 0.038 (0.043) 0.093** (0.040)Mother has a foreign nationality -0.216** (0.098) -0.133 (0.090)Live in publicy subsidized housing -0.089 (0.067) -0.144** (0.062)Child didn’t like maternal school 0.141** (0.067) 0.197** (0.094)Child didn’t learn a lot in maternal school -0.073 (0.069) -0.039 (0.076)Never fails due to excessive confidence -0.021 (0.036) 0.028 (0.036)Capable of regular attention 0.128** (0.055) 0.202*** (0.055)E"ciently completes tasks 0.150** (0.061) 0.005 (0.060)Never fatigues during school activities 0.015 (0.036) 0.075** (0.034)Consciously intervenes in class discussions 0.008 (0.050) -0.021 (0.048)Anticipates and is organized 0.088 (0.054) 0.108** (0.055)Requires support and encouragement -0.204*** (0.052) -0.205*** (0.052)Requires corrective measures and warning -0.173*** (0.055) -0.069 (0.073)Parent met with school psychologist -0.136 (0.100) -0.229** (0.110)Parent accompanied school excursion 0.043 (0.038) -0.066* (0.037)Parent picked up child after school -0.122*** (0.047) 0.013 (0.046)Parents regularly verify if child studies well -0.086** (0.039) 0.005 (0.037)Parents discuss class activities with child -0.017 (0.041) 0.091** (0.042)Parents check exercise book and schoolbag -0.022 (0.042) -0.065 (0.040)Child follows additional public coaching -0.356*** (0.123) -0.106 (0.142)Child follows additional private coaching -0.374** (0.148) -0.085 (0.125)Di"culties because child doesn’t listen well in class 0.038 (0.035) -0.056* (0.033)Child has di"culties because it’s hard -0.020 (0.034) -0.061* (0.032)Parents think child is good in writing 0.119*** (0.039) 0.155*** (0.044)Parents think child is good in mathematics 0.340*** (0.035) 0.290*** (0.035)Child does manual activities with family -0.131*** (0.040) -0.091** (0.039)Child plays parlor games with family member 0.065* (0.039) 0.132*** (0.037)Child reads a story with family member 0.001 (0.034) -0.071** (0.034)Child regularly watches TV before school -0.026 (0.036) -0.058 (0.038)Number of students in CE1 -0.006 (0.005) -0.012*** (0.005)Number of repeating students in CP 0.010 (0.013) -0.017 (0.014)School in priority education zone (ZEP) -0.189*** (0.071) -0.145** (0.073)Girl share in class 1998-1999 -0.185 (0.160) 0.031 (0.157)Knowledge of writing 0.045** (0.019) 0.037* (0.019)Verbal knowledge 0.066*** (0.022) 0.085*** (0.021)Reading 0.065*** (0.024) 0.078*** (0.021)Oral comprehension 0.029 (0.019) 0.025 (0.018)Temporal concepts 0.101*** (0.022) 0.167*** (0.023)Calculations 0.196*** (0.023) 0.172*** (0.021)Numbers and geometry 0.098*** (0.021) 0.024 (0.019)Observations 1871 1871R-squared 52.5% 50.7%


* p < .1, ** p < .05, *** p < .01

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The impact of school characteristics on achievement is typical. Girls in larger classes aswell as boys and girls in priority education zones obtain lower scores. Priority educationzones (“zones d’education prioritaire”) are areas which receive particular attention from theFrench government and often have smaller class sizes. Hence, the coe"cient on the numberof students per class indicates the impact of class size once the fact of studying in a priorityeducation zone is controlled for. Interestingly, the number of girls and boys in class does notmatter for mathematics achievement at the entry into 3rd grade. Still, but not statisticallysignificant, a higher share of girls negatively a!ects the outcome of boys.

Unsurprisingly, prior mathematics achievement strongly impacts later mathematics achieve-ment. Still, it is interesting to examine that the impact of prior achievement is smaller thanthe impact of some other explanatory variables such as the opinion of parents about theirchild’s achievement or the fact that the child follows additional coaching. Interestingly, theknowledge of general skills at the entry into 1st grade does not explain mathematics achieve-ment in 3rd grade, whereas prior French achievement does. Finally, the influence of historicalachievement measures on achievement in 3rd grade is similar for boys and girls.

Table 21 shows the results obtained by decomposing the gender gap for the average mathe-matics score as well as four sub-scores at the entry into 3rd grade into the components whichare due to explained and unexplained di!erences between boys and girls. The decompositionof both the explained and the unexplained components are split up into the seven previouslydescribed categories of explanatory variables. In the first section of Table 21, the averagepredicted scores for boys and girls as well as their di!erence are shown. Boys obtain on aver-age 0 for every score, given that all outcome variables were normalized for boys. As describedin Section 5, there is no gender di!erence on the average mathematics score. However, girlsscore almost 0.06 standard deviations better than boys in geometry, an advantage, whichis exactly compensated by boys in the evaluation of problem solving. As mentioned before,this contradicts the theory according to which boys have a particular advantage in exercicesdemanding spatial skills such as geometry.

Similar to what has been observed at the entry into primary school, gender di!erences inobservable characteristics predict an advantage for girls over boys in the average mathematicsscores, as well as in geometry and the evaluation of exercises. However, predicted genderdi!erences due to di!erences in characteristics are much smaller than at the entry into 1stgrade. This is due to the fact that most types of explanatory variables either do not di!eror do not su"ciently influence achievement in order to predict a di!erence. In particular,di!erences in maternal school inputs no longer lead to gender di!erences in mathematicsachievement, which may simply be explained with the fact that maternal school is su"cientlyfar away to have no influence on achievement in 3rd grade.

The only gender di!erence explainable with di!erent endowments results from di!erentnon-cognitive skills and achievement di!erences at the entry into 1st grade. Still, even con-sidering these two endowment di!erences, the predicted gender di!erence in average mathe-matics achievement is less than 0.05 standard deviations to the advantage of girls for geometryand exercises. For measuring and problem solving, di!erences in characteristics predict nogender di!erences at all. In conclusion, we observe that the gender gap can entirely be ex-plained with di!erences in observable characteristics for geometry and measuring. In exercisesand problem solving, endowment di!erences predict girls to score 0.05 standard deviationsabove their actual score.

The second half of Table 21 shows the unexplained part of the gender gap in mathematics,which stems from di!erences in the education production function of boys and girls. Similarto what was observed at the entry into 1st grade, but to a lesser extent and without beingstatistically significant, boys have higher returns to family inputs than girls.

An interesting observation concerns the unexplained part of the gender di!erence in geom-etry. According to di!erences in the coe"cients of the achievement production function other

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Table 21: Decomposition of gender gap into CE2

(1) (2) (3) (4) (5)Average Geometry Exercices Measuring Problems

Prediction for boys -0.000 -0.000 -0.000 0.000 0.000(0.023) (0.023) (0.023) (0.023) (0.023)

Prediction for girls -0.011 0.057** -0.036* 0.005 -0.058***(0.022) (0.023) (0.022) (0.023) (0.022)

Di!erence 0.011 -0.057* 0.036 -0.005 0.058*(0.032) (0.032) (0.032) (0.033) (0.032)

Explained gender di!erencesFamily input before school -0.001 0.002 -0.002 0.000 -0.001

(0.004) (0.004) (0.004) (0.003) (0.003)Maternal school input 0.004 0.006 0.001 0.005 0.005

(0.003) (0.004) (0.003) (0.004) (0.004)Noncognitive skills -0.035*** -0.032*** -0.031*** -0.029*** -0.020**

(0.009) (0.009) (0.009) (0.009) (0.008)Family school contact -0.003 -0.001 -0.003 -0.003 -0.002

(0.002) (0.002) (0.002) (0.003) (0.003)Family input during school 0.025 0.008 0.025 0.020 0.032

(0.018) (0.021) (0.020) (0.020) (0.021)Primary school input 0.004 0.003 0.001 0.006 0.004

(0.007) (0.008) (0.007) (0.008) (0.008)Achievement at entry into school -0.030* -0.034** -0.039*** -0.006 -0.008

(0.016) (0.015) (0.015) (0.015) (0.015)Total -0.035 -0.048 -0.047 -0.006 0.009

(0.031) (0.031) (0.031) (0.031) (0.031)

Unexplained gender di!erencesFamily input before school 0.106 0.101 0.047 0.058 0.249

(0.142) (0.168) (0.158) (0.164) (0.166)Maternal school input 0.000 0.019 -0.001 -0.012 0.001

(0.015) (0.018) (0.017) (0.018) (0.018)Noncognitive skills -0.082 0.206* -0.140 -0.152 -0.120

(0.104) (0.123) (0.116) (0.120) (0.121)Family school contact -0.088 0.004 -0.049 -0.147** -0.120*

(0.058) (0.069) (0.065) (0.067) (0.068)Family input during school -0.036 -0.022 -0.041 -0.004 -0.061

(0.099) (0.117) (0.110) (0.114) (0.115)Primary school input 0.013 0.113 0.060 -0.205 0.131

(0.181) (0.214) (0.201) (0.209) (0.211)Achievement at entry into school -0.003 0.032** -0.005 -0.024* -0.011

(0.012) (0.014) (0.013) (0.013) (0.013)Intercept 0.136 -0.461 0.212 0.487 -0.020

(0.277) (0.327) (0.307) (0.319) (0.323)Total 0.046 -0.009 0.084** 0.001 0.049

(0.031) (0.037) (0.034) (0.035) (0.036)


* p < .1, ** p < .05, *** p < .01

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than the intercept, boys are predicted to outperform girls by about 0.45 standard deviations.In particular, for the same level of non-cognitive skills, boys score of 0.2 standard deviationshigher than girls. Most other inputs also predict higher returns for boys than for girls. How-ever, girls entirely compensate these higher returns with the intercept. In total, di!erencesin the achievement production function therefore do not result in gender di!erences. Forthe geometry score overall, this can be interpreted as follows. Endowment di!erences predictgirls to score 0.05 standard deviations higher. However, boys are more able to transform theirendowments into achievement, a fact which is compensated by an unexplainable advantagefor girls. Hence, on average, the achievement production function makes no di!erence andgender di!erences can be entirely explained by di!erent inputs.

While the advantage of girls in geometry is entirely explainable with endowment di!er-ences, boys outperform girls in problem solving exclusively due to structural di!erences. Inparticular, boys benefit from higher returns to family input before school and school input.The intercept for problem solving is 0, which means that there is no entirely unexplainablegender di!erence.

There is no gender gap in the sub-scores for exercises and measuring. For the former, theadvantage of girls resulting from better endowments is entirely compensated by an unex-plainable advantage for boys stemming from a higher intercept. No gender di!erence, neitherin endowments, neither in the structure, can be observed for measuring. Here, the only re-markable fact is that boys have lower returns to inputs, but compensate them with a higherunexplainable score due to the intercept.

6.3 Evaluation at the entry into 6th grade

While the gender gap in mathematics was either small or inexistent, it is considerably greaterin 6th grade (sixieme). Table 22 presents the coe"cients of the OLS for boys and girls,including all relevant family and school inputs as well as prior achievement and non-cognitiveskills. Here, the dependent variable is the normalized average mathematics score at the entryinto 6th grade. Some of the variables that were included in the regression are omitted in thetable in order to improve readability.

The impact of socio-economic status on mathematics achievement in 6th grade is typical.Interestingly, having a mother of foreign nationality seems to be positively correlated withmathematics achievement, at least for girls this is statistically significant. This is surprisingas foreign nationality is often associated with weaker achievement. A possible explanationcould be that students with immigrant parents from a disadvantaged background have left thesample, which leaves only those students with foreign parents, who come from an advantagedbackground, such as parents from other European countries or educated immigrants fromNorth Africa. Similar to CE2, parental employment no longer has an e!ect on achievement.Variables measured many years earlier have a relatively weak impact. This is particularlytrue for measures of non-cognitive skills, the impact of which declines and even becomescounterintuitive especially for boys. This should not be of great concern, given that theywere measured five years prior to the entry into 6th grade. In this context, it is remarkablethat the opinion of parents on their child’s maternal school achievement still matters for theachievement of boys in 6th grade.

Variables indicating parents’ involvement into school activities have expected and signifi-cant e!ects. In particular the fact that parents have met the school psychologist and that thechild takes additional private coaching are good predictors for low mathematics achievement.Interestingly, parents’ opinion on their child’s mathematics (and even reading) ability – basedon questions which were asked in 2nd grade – still have an important impact on 6th grademathematics achievement. Home activities influence the outcome only to a very small extent.

Several achievement measures, which stem from the evaluation of cognitive skills at theentry into 1st grade, are correlated with mathematics achievement in 6th grade. Interestingly,

53

Table 22: Mathematics achievement at the entry into 6th grade

Boys GirlsMother works -0.036 (0.047) 0.002 (0.047)Socio-professional category 3 or 4 (1997, father) 0.084** (0.041) -0.012 (0.042)Mother has obtained university diploma 0.112** (0.044) 0.166*** (0.048)Mother has less than vocational school diploma -0.189*** (0.052) -0.049 (0.053)Mother has a foreign nationality 0.083 (0.115) 0.310*** (0.120)Live in publicy subsidized housing 0.016 (0.078) -0.090 (0.076)More than 3 years maternal school attendance 0.014 (0.041) -0.069 (0.042)Child didn’t learn a lot in maternal school -0.155* (0.080) 0.012 (0.073)Self-confident during school activities -0.043 (0.057) -0.014 (0.057)Never fails due to excessive confidence 0.079** (0.039) -0.000 (0.042)Autonomous 0.002 (0.062) -0.127* (0.070)Never fatigues during school activities -0.076* (0.039) -0.038 (0.041)Anticipates and is organized -0.019 (0.056) 0.081 (0.066)Requires support and encouragement -0.094 (0.060) -0.048 (0.064)Requires corrective measures and warning -0.025 (0.064) -0.191** (0.088)Parent met with school psychologist -0.210* (0.122) -0.282*** (0.106)Parent picked up child after school 0.029 (0.052) -0.009 (0.051)Parents regularly verify if child studies well -0.045 (0.043) -0.098** (0.042)Parents check exercise book and schoolbag -0.075 (0.047) 0.009 (0.045)Child follows additional private coaching -0.197 (0.163) -0.399*** (0.149)Parents think child doesn’t listen well in class 0.023 (0.036) 0.010 (0.038)Parents think child is good in writing 0.033 (0.048) 0.100* (0.056)Parents think child is good in mathematics 0.171*** (0.039) 0.229*** (0.040)Child reads a story with family member -0.031 (0.039) 0.041 (0.040)Child enrolled in sports club 0.038 (0.044) 0.081** (0.040)Child enrolled in music school 0.045 (0.051) 0.036 (0.039)Child regularly watches TV before school 0.041 (0.040) -0.038 (0.043)Technical knowledge (CP) -0.012 (0.022) -0.027 (0.020)Knowledge of writing (CP) 0.022 (0.021) 0.048** (0.023)Verbal knowledge (CP) 0.017 (0.025) 0.065** (0.027)Spatial skills (CP) -0.013 (0.019) -0.021 (0.021)Temporal concepts (CP) 0.058** (0.023) 0.094*** (0.028)Calculations (CP) 0.114*** (0.028) 0.093*** (0.026)Numbers and geometry (CP) 0.026 (0.024) -0.014 (0.022)Number of students in CM2 -0.008 (0.005) 0.010** (0.005)Number of foreign students in CM2 0.010 (0.012) -0.005 (0.012)School in priority education zone (ZEP) in CM2 -0.063 (0.103) -0.248** (0.104)Child went to private school in CM2 -0.073 (0.050) -0.040 (0.054)City with more than 100,000 people 0.188*** (0.043) 0.028 (0.045)Girl share in CE2 -0.092 (0.176) 0.230 (0.167)Geometry (CE2) 0.094*** (0.024) 0.069** (0.027)Exercices (CE2) 0.187*** (0.030) 0.151*** (0.030)Measuring (CE2) 0.060** (0.027) 0.114*** (0.027)Problems (CE2) 0.075*** (0.025) 0.165*** (0.028)Comprehension (CE2) 0.106*** (0.031) 0.083*** (0.031)Language tools (CE2) 0.072*** (0.028) 0.016 (0.032)Writing (CE2) 0.001 (0.025) 0.049* (0.027)Observations 1426 1436R-squared 57.6% 56.6%

Standard errors in parentheses * p < .1, ** p < .05, *** p < .01

54

and unlike the theories cited earlier (Niederle and Vesterlund 2010), spatial skills, as measuredat the entry into 1st grade, do not seem to influence mathematics achievement.

School inputs from the first two years of primary school were removed from the regression,given that the same information (number of students, number of foreign students, priorityeducation zone etc.) was also available for more recent years. Moreover, it is not usefulto include the same statistics for multiple years, as all school input variables are stronglycorrelated between one year and another. Interestingly, the fact of being enrolled in a schoolsituated in a priority education zone does not influence the mathematics score of boys. Theshare of girls in the student’s class has a positive impact for the achievement of girls and anegative on boys, however, both are not statistically significant.

Achievement in 3rd grade is a strong predictor for achievement in 6th grade. Interest-ingly, and again contrary to the theory highlighted among others by Niederle and Vesterlund(2010), prior achievement in geometry only very weakly influences later average mathematicsachievement.

Tables 23 and 24 decompose the gender gap for all ten sub-scores of mathematics achieve-ment. Both tables show the decomposition of the average score as a reference. Table 23decomposes the gender gap for the five sub-scores evaluating the student’s skills in five the-matic fields: analyzing a situation (model 2), applying a technique (model 3), geometry(model 4), numerical problems (model 5) and applying operators (model 6). In Table 24, wesee the decomposition for the scores evaluating the knowledge of methods used in mathemat-ics such as problem solving (model 2), providing an answer (model 3), researching information(model 4), treating information (model 5) and using knowledge (model 6). Please refer toSection 4 for more details on the di!erent scores.

In the first part of Table 23 we see the average gender gap for each subscore. We cansee that on average girls score 0.19 standard deviations below boys. Note that the averagegender gap is smaller here than in Section 5. This is the unfortunate consequence of missingdata, which obliges us to restrict the analysis to the number of individuals for whom allvariables are available. When examining the thematic sub-scores of Table 23, we observethat girls score particularly low compared to boys in exercises examining their capacity totreat information as well as to solve numerical exercises. The gender gap is particularly weakin geometry and problem solving.

Very little of the gender gap in mathematics at the entry into 6th grade can be explainedwith di!erences in observable characteristics, as the section entitled “Explained gender dif-ferences” of Table 23 shows. Di!erences in endowments would predict an low advantagefor girls of approximately 0.03 to 0.05 standard deviations, depending on the score. Thepredicted advantage for girls is – similar to what we examined in CE2 – particularly due todi!erences in non-cognitive skills as well as di!erences in achievement at the entry into CE2.Even though their evaluation dates five years back, di!erences in non-cognitive skills predicta small but significant advantage of 0.02 standard deviations for girls in treating information,numerical problems and evaluations involving the use of operators, as well as for the averagescore. Di!erences in achievement at the entry into CE2 predict a significant advantage ofapproximately 0.05 standard deviations for girls in geometry as well as in exercises involvingthe use of operators. Interestingly, di!erences in family input during primary school predictan advantage of 0.04 standard deviations for boys in the treatment of operators. Overall,these explainable gender di!erences are small in magnitude.

Unlike what we observed for previous years, the gender gap at the entry into 6th gradecannot be explained with higher returns to inputs for boys. On the contrary, di!erences inthe educational achievement production function between genders predict strong advantagesfor girls resulting from higher returns to family input during primary school as well as higherreturns to school input. Di!erences in the coe"cients relating school input in the last twoyears of primary school to mathematics achievement at the entry into 6th grade predict an

55

Tab

le23

:Decom

positionof

gender

gapat

theentryinto

6thgrad

e(6e)

–Part1(K

now

ledge

ofmathem

atical

topics)

(1)

(2)

(3)

(4)

(5)

(6)

Average

Problems

Treat

info

Geometry

Numbers

Operators

Prediction

forboy

s0.00

0(0.027

)0.00

0(0.027

)-0.000

(0.027

)0.00

0(0.027

)-0.000

(0.027

)0.00

0(0.027

)Prediction

forgirls

-0.187

***(0.02

7)-0.052

*(0.028

)-0.312

***(0.02

8)-0.011

(0.027

)-0.257

***(0.02

8)-0.126

***(0.02

6)Di!eren

ce0.18

7***

(0.038

)0.05

2(0.039

)0.31

2***

(0.039

)0.01

1(0.038

)0.25

7***

(0.039

)0.12

6***

(0.037

)

Explained

genderdi!eren

ces

Fam

ilyinputbeforeschoo

l-0.000

(0.005

)0.00

0(0.004

)-0.000

(0.004

)0.00

0(0.006

)-0.001

(0.005

)-0.001

(0.005

)Maternal

schoo

linput

-0.001

(0.004

)-0.005

(0.005

)0.00

1(0.004

)0.00

2(0.004

)-0.003

(0.004

)-0.001

(0.004

)Non

cogn

itiveskills

-0.016

**(0.008

)0.00

8(0.009

)-0.015

*(0.008

)-0.013

(0.009

)-0.021

**(0.009

)-0.015

*(0.009

)Fam

ilyschoo

lcontact

-0.003

(0.003

)-0.005

(0.003

)-0.004

(0.003

)-0.004

(0.003

)0.00

1(0.003

)-0.003

(0.003

)Fam

ilyinputduringschoo

l0.01

6(0.019

)-0.025

(0.023

)0.01

9(0.022

)0.00

5(0.023

)0.00

7(0.022

)0.04

3**(0.021

)Achievementat

schoo

lentry

-0.008

(0.012

)-0.016

(0.014

)-0.010

(0.013

)-0.004

(0.013

)-0.005

(0.013

)-0.005

(0.013

)Schoo

linputCM1&

CM2

-0.001

(0.008

)0.00

9(0.009

)-0.003

(0.008

)0.01

2(0.009

)-0.006

(0.009

)-0.013

(0.009

)Achievementin

CE2

-0.032

(0.021

)-0.014

(0.020

)-0.021

(0.020

)-0.045

**(0.019

)-0.006

(0.020

)-0.033

*(0.020

)Total

-0.045

(0.037

)-0.048

(0.037

)-0.033

(0.037

)-0.046

(0.037

)-0.033

(0.037

)-0.028

(0.037

)

Unexplained

genderdi!eren

ces

Fam

ilyinputbeforeschoo

l-0.118

(0.169

)0.04

1(0.210

)-0.283

(0.196

)0.02

2(0.199

)-0.077

(0.199

)-0.146

(0.190

)Maternal

schoo

linput

0.00

9(0.017

)0.02

4(0.021

)0.02

3(0.020

)-0.000

(0.020

)0.00

1(0.020

)-0.000

(0.019

)Non

cogn

itiveskills

0.16

3(0.120

)0.11

3(0.150

)0.12

8(0.140

)0.07

6(0.142

)0.23

1(0.142

)0.12

5(0.136

)Fam

ilyschoo

lcontact

0.03

6(0.066

)0.00

1(0.082

)0.01

3(0.077

)0.00

6(0.078

)0.05

7(0.078

)0.06

2(0.075

)Fam

ilyinputduringschoo

l-0.126

(0.112

)-0.311

**(0.140

)-0.068

(0.130

)-0.082

(0.133

)-0.094

(0.132

)-0.071

(0.127

)Achievementat

schoo

lentry

-0.018

(0.014

)0.02

7(0.017

)-0.018

(0.016

)-0.019

(0.016

)-0.025

(0.016

)-0.018

(0.016

)Schoo

linputCM1&

CM2

-0.525

**(0.215

)-0.680

**(0.267

)-0.227

(0.249

)-0.893

***(0.25

3)-0.029

(0.253

)-0.343

(0.242

)Achievementin

CE2

0.01

5(0.017

)-0.007

(0.022

)-0.008

(0.020

)0.03

6*(0.020

)0.01

2(0.020

)0.01

2(0.019

)Intercep

t0.79

7**(0.322

)0.89

2**(0.401

)0.78

6**(0.374

)0.91

2**(0.380

)0.21

4(0.380

)0.53

3(0.363

)Total

0.23

2***

(0.036

)0.09

9**(0.045

)0.34

5***

(0.041

)0.05

7(0.043

)0.29

0***

(0.042

)0.15

4***

(0.040

)Observations

2862

2862

2862

2862

2862

2862

Standarderrors

inparen

theses

*p<

.1,**

p<

.05,

***p<

.01

56

advantage for girls of almost 0.7 standard deviations for problem solving exercises. In geom-etry, girls are even predicted to score almost 0.9 standard deviations higher than boys due tohigher returns to school inputs. Therefore, di!erences in the returns to school input predictan advantage of more than half a standard deviation for girls on the average mathematicsscore. Moreover, higher returns to family input during primary school predicts an additionaladvantage of 0.31 standard deviations for girls in problem solving.

From the di!erences in returns to school and family input, the score of girls in problemsolving as well as geometry is thus predicted to be about 1 standard deviation higher thanthe score of boys. Yet, we observe a slight disadvantage for girls in both scores. The dif-ference results from an intercept in the educational achievement production function, whichis approximately 0.9 standard deviations higher for boys than for girls. Hence, boys scorehigher than girls due to a parameter which is entirely unexplainable. The only score wherethe advantage of boys is not entirely unexplainable is the evaluation of numerical exercises.Here, higher returns to non-cognitive skills explain more than 0.2 standard deviations of thedi!erence, however, this number is at the limit of being statistically significant.

A similar situation can be observed in Table 24, which shows the subscores evaluating theknowledge of five methodologies used in mathematics. On average, girls score approximately0.2 standard deviations below boys in the evaluation asking them to analyze a situation,research information and use knowledge. Girls score only slightly less than boys when askedto produce an answer. In the evaluation requiring students to apply a technique, girls scoreon average even 0.1 standard deviations higher than boys. With these figures, the genderdi!erence is slightly weaker than the gap described in Section 5, which is due to the fact thathere the analysis is restricted to students for whom each variable was available.

The predictions resulting from di!erences in endowments are similar to those in the previoustable. Di!erences in non-cognitive skills and achievement in CE2 predict an advantage forgirls of approximately 0.05 standard deviations for all scores except the analysis of a situation.For the latter, no gender gap is predicted. In addition, girls seem to benefit from higherreturns to most inputs. Di!erences in returns to school inputs to the advantage of girls areparticularly strong in the evaluations testing the students’ capacity to analyze a situation,produce an answer and use knowledge.

In the evaluation asking students to analyze a situation, the predicted advantage for girlsresulting from higher returns to school inputs is even increased, because returns are equallyhigher to family input during school. However, the achievement production function for boyscompensates the predicted advantage for girls with an intercept leading boys to score morethan 1 standard deviation above girls.

In the both the evaluation of the students’ capacity to use knowledge as well as the oneassessing their ability to produce an answer, the above-mentioned predicted advantage forgirls is more than compensated by boys with higher returns to non-cognitive skills as well aswith the intercept. The returns to non-cognitive skills are especially high for boys when itcomes to producing an answer, where they score almost 0.33 standard deviations above girlsfor the same level of non-cognitive skills. The only score for which boys do not compensatetheir predicted disadvantage with structural di!erences in the education production functionis the evaluation of applying a technique and thus the only score where girls score better thanboys.

57

Tab

le24

:Decom

positionof

gender

gapat

theentryinto

6thgrad

e(6e)

–Part2(K

now

ledge

ofmethod

susedin

mathem

atics)

(1)

(2)

(3)

(4)

(5)

(6)

Average

Situation

Answ

erResearch

Technique

Know

ledge

Prediction

forboy

s0.00

0(0.027

)-0.000

(0.027

)0.00

0(0.027

)0.00

0(0.027

)-0.000

(0.027

)0.00

0(0.027

)Prediction

forgirls

-0.187

***(0.02

7)-0.217

***(0.02

7)-0.031

(0.027

)-0.164

***(0.02

8)0.09

8***

(0.026

)-0.211

***(0.02

7)Di!eren

ce0.18

7***

(0.038

)0.21

7***

(0.038

)0.03

1(0.038

)0.16

4***

(0.039

)-0.098

***(0.03

7)0.21

1***

(0.038

)

Explained

genderdi!eren

ces

Fam

ilyinputbeforeschoo

l-0.000

(0.005

)0.00

0(0.004

)-0.001

(0.005

)-0.000

(0.005

)0.00

0(0.004

)-0.001

(0.005

)Maternal

schoo

linput

-0.001

(0.004

)-0.000

(0.004

)-0.001

(0.004

)0.00

0(0.005

)-0.007

(0.005

)0.00

1(0.004

)Non

cogn

itiveskills

-0.016

**(0.008

)-0.009

(0.008

)-0.022

**(0.009

)-0.012

(0.009

)-0.011

(0.010

)-0.015

*(0.008

)Fam

ilyschoo

lcontact

-0.003

(0.003

)-0.003

(0.003

)-0.008

**(0.004

)-0.003

(0.004

)-0.002

(0.003

)0.00

0(0.003

)Fam

ilyinputduringschoo

l0.01

6(0.019

)0.03

3(0.021

)0.00

0(0.023

)-0.001

(0.023

)0.03

0(0.026

)-0.004

(0.021

)Achievementat

schoo

lentry

-0.008

(0.012

)-0.017

(0.012

)-0.016

(0.013

)-0.006

(0.013

)0.01

4(0.014

)0.00

5(0.013

)Schoo

linputCM1&

CM2

-0.001

(0.008

)0.00

7(0.008

)-0.003

(0.009

)0.00

1(0.009

)-0.014

(0.010

)-0.006

(0.008

)Achievementin

CE2

-0.032

(0.021

)-0.024

(0.021

)-0.014

(0.019

)-0.024

(0.020

)-0.058

***(0.01

9)-0.026

(0.019

)Total

-0.045

(0.037

)-0.013

(0.037

)-0.065

*(0.037

)-0.045

(0.037

)-0.049

(0.038

)-0.047

(0.037

)

Unexplained

genderdi!eren

ces

Fam

ilyinputbeforeschoo

l-0.118

(0.169

)-0.063

(0.185

)0.03

5(0.202

)-0.226

(0.210

)-0.217

(0.228

)-0.129

(0.185

)Maternal

schoo

linput

0.00

9(0.017

)0.00

0(0.019

)0.01

5(0.021

)0.02

9(0.021

)-0.019

(0.023

)0.01

1(0.019

)Non

cogn

itiveskills

0.16

3(0.120

)0.02

6(0.131

)0.32

7**(0.144

)0.19

5(0.150

)0.04

6(0.162

)0.17

3(0.132

)Fam

ilyschoo

lcontact

0.03

6(0.066

)0.00

2(0.072

)0.01

7(0.079

)0.05

9(0.083

)0.00

4(0.090

)0.06

9(0.073

)Fam

ilyinputduringschoo

l-0.126

(0.112

)-0.269

**(0.123

)-0.075

(0.134

)0.00

7(0.140

)0.08

2(0.152

)-0.003

(0.124

)Achievementat

schoo

lentry

-0.018

(0.014

)-0.007

(0.015

)-0.005

(0.017

)-0.007

(0.018

)0.00

8(0.019

)-0.042

***(0.01

5)Schoo

linputCM1&

CM2

-0.525

**(0.215

)-0.560

**(0.235

)-0.596

**(0.256

)-0.237

(0.267

)0.06

8(0.290

)-0.404

*(0.235

)Achievementin

CE2

0.01

5(0.017

)0.02

9(0.019

)-0.011

(0.021

)-0.016

(0.021

)0.02

4(0.023

)0.01

1(0.019

)Intercep

t0.79

7**(0.322

)1.07

1***

(0.352

)0.38

9(0.385

)0.40

3(0.402

)-0.044

(0.436

)0.57

2(0.354

)Total

0.23

2***

(0.036

)0.22

9***

(0.039

)0.09

6**(0.043

)0.20

9***

(0.044

)-0.050

(0.049

)0.25

8***

(0.040

)Observations

2862

2862

2862

2862

2862

2862

Standarderrors

inparen

theses

*p<

.1,**

p<

.05,

***p<

.01

58

7 Conclusion: Discussion and limits

The findings of this paper can be summarized as follows. While a gender gap in French alreadyexists, there is no gender gap in mathematics at the entry into primary school. At the entryinto 3rd grade, still no gender gap exists for the average mathematics achievement. However,girls score slightly better than boys in geometry and slightly weaker in numerical exercisesand problem solving. A considerable gender gap in mathematics can for the first time beobserved at the entry into 6th grade. On average, girls score almost a quarter of a standarddeviation lower than boys. Gender di!erences vary strongly between sub-scores. They areparticularly strong for treating information and analyzing a situation and particularly weakin geometry, numerical problems and applying a technique.

Gender di!erences throughout the distribution reverse between 3rd and 6th grade. Whilegirls at the lower end of the distribution score just as good or even better than boys in 3rdgrade, they are particularly weak in 6th grade. At the upper end of the distribution, thegender di!erence is small in 6th grade, while it is strongest in 3rd grade. With other words,a small gender di!erence already exists at the upper end of the distribution in 3rd gradeand does not aggravate in 6th grade. At the lower end of the distribution, however, girlsobtain the same scores as boys in 3rd grade, whereas they score about a third of a standarddeviation lower than boys in 6th grade. Hence, the gender gap seems to develop earlier amonghigh-achieving students.

The decomposition of the gender gap into endowment and structural di!erences revealsthe following pattern. Di!erences in observable characteristics predict girls to score higherthan boys. Boys compensate the predicted weakness due to structural di!erences in theeducational achievement production function. In 1st and 3rd grade, boys compensate thepredicted disadvantage with higher returns to the same level of characteristics. In 6th grade,returns to school input are considerably higher for girls, a further disadvantage which boyscompensate with an entirely unexplainable component.

Concerning the theories mentioned before, we can make the following observations. Theoften cited theory according to which gender di!erences in mathematics can be explainedwith higher achievement of boys in spatial skills and geometry cannot be confirmed. First,boys do not outperform girls in spatial skills, at least not in the way they are measuredat the entry into primary school. Second, unlike the prediction of the theory, boys are notparticularly strong in geometry. On the contrary, in 6th grade, girls only score slightly lowerand in 3rd grade even higher than boys. Third, geometry and spatial skills do not evenseem to play an important role in predicting future mathematics achievement. The averagemathematics score in 6th grade cannot be predicted at all with the score on spatial skills andonly very weakly with the results obtained in geometry three years earlier.

Another theory which is often mentioned (see Section 2) explains the gender gap with theexistence of stereotypes. If stereotypes are strongly present among parents and teachers,girls might score weaker in mathematics simply because they are educated to think that they

Table 25: Parents’ opinion on the mathematics ability of their child

Gender of studentIn mathematics, my child is... male female TotalGood 35% 26% 31%Rather good 50% 55% 53%Rather weak 13% 17% 15%Weak 2% 2% 2%Notes: Numbers indicate the percentage of boys’ and girls’ parents giving eachpossible answer. Questions were asked in 1999 (when child was in 2nd grade).

59

should concentrate on language skills rather than mathematics. Indeed, the parent question-naire in the French Primary School Panel reveals stereotypical opinions among parents. Asan example, Table 25 shows the distribution of opinions of parents regarding the mathematicsachievement of their child by gender. We can see that the parents of boys are much morelikely to think that their child is good in mathematics. This is even more astonishing, as atthe time the parent questionnaire was administered, no gender gap in mathematics achieve-ment could yet be observed. The regression estimates presented in Section 6 also show that apositive opinion of parents on their child’s mathematics achievement is correlated with betteractual achievement. However, we cannot be certain about the interpretation, as opinion andactual achievement can be correlated in both directions. Either, parents have a good opinionbecause their child obtains good marks. Or, a child is good in mathematics because her orhis parents encourage her to be so by having a positive opinion. Only in the latter case, wecan defend the theory of stereotypes.

A third possible explanation of the gender gap relates achievement to di!erential treatmentby teachers. This theory is very related to stereotypes and states that teachers might providebetter support to boys rather than girls when it comes to mathematics. This could be becauseteachers have the stereotypical opinion according to which boys should be particularly goodin mathematics. Girls could also score weaker because they are particularly stimulated if theirteacher is female, whereas mathematics teachers are often male. It is di"cult to verify anyof these theories in detail with the data we have. A potential hint for stereotypical thinkingof teachers could be seen in the fact that teachers systematically give better scores for girlsin the evaluation of non-cognitive skills. However, it is impossible to distinguish betweenthe actual level of non-cognitive skills of the student and the teacher’s subjective assessmentwhich might be led by stereotypical thinking.

This study sheds some light on the extent and the determinants of the mathematics gendergap in French primary school. However, several issues must be highlighted which limit thevalidity of the results found in this paper. The limits concern both the available data andthe assumptions of the model relating inputs to educational achievement.

First, the French Primary School Panel is strongly subject to attrition. Many studentshave left the sample and could not be followed into their new school. While more than 9200students participated in the evaluation at the entry into 1st grade, less than 8000 took partin the test at the entry into 3rd and 6th grade. Students from disadvantaged backgroundsare more likely to leave the sample than students from a more favored background. Forexample, among all students, 25% of the mothers obtained a university degree, whereas 29%of the mothers are university graduates among students who took part in the evaluation atthe entry into 6th grade. However, we have no reason to believe that girls quit the panelmore or less often than boys. Hence, attrition does not distort the balance between boys andgirls, but truncates the distribution which is available for our analysis. The French PrimarySchool Panel contains weights for each student for each year, which probably aim to correctattrition. However, I did not use these weights given that their meaning was unclear andgiven that girls and boys have the same propensity to leave the sample.

A more serious problem is the fact that many students in French primary school are obligedto repeat a class. Indeed, 6% of students repeated either of the first two school years andalmost 20% of the students took the evaluation at the entry into 6th grade one year laterthan normal4. In this paper, I ignore those students who took the test one year later. This isproblematic, as boys are more likely to repeat a class than girls. At the entry into 1st grade,when all students are still in the same class, the girl share of the sample is 49%. Among thestudents who do the examination at the entry into 6th grade one year later, 58% are boys.This leads to a girl share of 50.8% in the sample of students who took the test in the usualyear.

41693 out of 9641 students took the test at the entry into 6th grade in 2003 rather than in 2002.

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Third, the French Primary School Panel contains many missing values. Variables arenot missing at random, but are more likely to miss for students from a less favored familybackground. This is due to the fact that these students are more often absent or that it is moredi"cult to reach their parents. We see the consequences of this when comparing the figuresabout the gender gap found in Sections 5 and 6. In Section 5, I describe the gender gap amongall students (except repeaters) for whom achievement scores were available. This concerns7321 students in 3rd and 6401 students in 6th grade. The decomposition in Section 6 can,of course, only be carried out for students, for whom information for each of the explanatoryvariables is available, which strongly reduces the sample to 3742 students in 3rd and 2862students in 6th grade. While the gender gap on the average mathematics score in 6th grade is0.24 standard deviations with the larger sample used in Section 5, it reduces to 0.19 standarddeviations when restricting the sample to the students with all explanatory variables, asdone in Section 6. The smaller gap with the smaller sample confirms the theory that weakerstudents leave the sample, given that the gap is largest among weak students.

Finally, the identifying assumptions made in order for the education production model aswell as the decomposition to be valid may be false. In particular, we need to assume thatunobserved characteristics are not correlated with group membership or observed characteris-tics once the scores of the non-cognitive skill evaluation are included as explanatory variables.This is possibly wrong, given that the non-cognitive skill evaluation only imperfectly capturesability.

In line with recent research by Fryer and Levitt (2010) and Sohn (2010), I find that thegender gap in mathematics, which arises in primary school, is large and persistent in Francetoo. The results of this paper can be considered as a first contribution of a quantitativedescription of the gender gap in French primary school and could be extended towards moredetail. It might be interesting to examine the components contributing to the evolution ofthe gap between 3rd and 6th grade. Moreover, more attention should be devoted towardsthe 20% of students who repeat a class in primary school. Given that the gender gap wasfound to be strongest at the bottom of the distribution, understanding gender di!erences ofachievement among repeaters will strongly contribute to a more thorough understanding ofthe subject. Finally, the examination of school characteristics, in particular the share of girlsand boys in the student’s class, might help further understand the role of stereotypes in thedetermination of the gender gap.

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