German undergraduate mathematics enrolment

International Journal of Mathematical Education in Science and Technology, Vol. 41, No. 4, 15 June 2010, 435–449

German undergraduate mathematics enrolment numbers: background and change Claudia Ammanna, Jo¨rg Frauendienerab and Derek Holtona* a

Department of Mathematics and Statistics, University of Otago, Box 56, Dunedin, New Zealand; bCentre of Mathematics for Applications, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway

Downloaded by [University of Otago] at 19:18 16 June 2014

(Received 3 March 2009) Before we consider the German tertiary system, we review the education system and consider other relevant background details. We then concentrate on the tertiary system and observe that the mathematical enrolments are keeping up with the overall student enrolments. At the same time, the first year mathematics enrolments for women are greater than that for men, although more men are still studying mathematics at university. Finally, we note that the German economy seems to play a role in mathematics enrolments though not necessarily to its comparative detriment. Keywords: recruitment; retention; enrolment

1. Introduction This article arose as a consequence of the work of Survey Team 1 for ICME 10 in Monterrey, Mexico (‘Recruitment, entrance and retention of students to university mathematics studies in different countries’). The work of the Survey Team led to a special issue of iJMEST [1]. This article is complementary to many of those in that special issue. Here, we are trying to determine what effects there are on the enrolment of students in undergraduate mathematics courses in Germany. We see that currently mathematics is performing well against overall enrolments and that the proportion of students enrolling for mathematics relative to all subjects is increasing slightly. There also appears to be an inverse relationship between the state of the economy and mathematics enrolments as well as overall enrolments. But, we also take the opportunity in this article to present a quick overview of the German educational system from primary school to show the pathways that exist for German students. At this point we note that by ‘mathematics enrolments’ we mean the number of students at different universities that are enrolled in mathematics as their subject of interest – what might be called in some countries, the number of mathematical majors. This number includes teachers who intend to teach mathematics, but it does not include students in other subject areas who may be required to take some mathematics courses. Hence, physicists and engineers are not included in the number of mathematics enrolments that we quote here. *Corresponding author. Email: [email protected] ISSN 0020–739X print/ISSN 1464–5211 online ! 2010 Taylor & Francis DOI: 10.1080/00207390903564629 http://www.informaworld.com

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2. Primary education


In Germany, education is under the control of each state. The Federal states set their own curriculum but are bound by the curriculum framework of the whole federation. So, under the constitution, the Federal states have to ensure equal educational structures and equal achievements. We will give a snapshot of the main sections of school and tertiary education based on the model provided by the Federal Ministry of Education and Research (Figure 1) [2–4]1 and details provided by the States of Bayern (available

Figure 1. The German education system.

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at http://www.stmuk.bayern.de/km/education/overview/), Baden-Wu¨rttemberg and Nordrhein-Westfalen [5]. Details of any particular state can be found by searching their website. Informal education begins for children from 3 years of age in Kindergarten, which aim to establish an initial learning experience for children. Kindergarten is run both by the states and by local private, non-profit making bodies, which are generally churches and charities. However, all of these institutions are subject to established state standards [3]. Children attend Grundschule (primary school) from age 6 to 10. This covers grades 1–4. As in most countries, German children at this stage are in classes of approximately 22 students with a single teacher compared with 25 in secondary school [6]. However, access to a foreign language, usually English or French, begins as early as grade 1.

3. Secondary education The three types of secondary schools in Germany, going from academic to practically oriented, are the Gymnasium (grammar school), the Realschule (intermediate school) and the Hauptschule (secondary general school) (Figure 1). General education in Germany is compulsory from the year children turn 6 until they are 15 or 16. At that point, they may choose between employment, an apprenticeship or further study. At the end of grade 4, students go to whichever of these three types of schools best fits their ability and aspirations. So children are split by aptitude as early as grade 4. It is possible to change direction, but this does mean that they need to have good results if they wish to move to a higher academically oriented school. In some Federal states (e.g. Nordrhein-Westfalen), the fifth grade (and sometimes also the sixth grade) is offered as an orientation stage, after which parents and teacher decide whether the child continues with that selected school type or not. In grades 5–10 of the Gymnasium, there is a single set course for all students. Only in the last 2 years of the Gymnasium do students have choices and these depend more on their areas of interest than on the particular form of the Abitur that they plan to take. It is necessary for students to take three main areas in that last 2 years of their education: (1) languages, literature, art; (2) social sciences and (3) mathematics, natural sciences and techniques [7]. There are two levels of mathematics in the last 2 years at Gymnasium open to students studying for the Abitur – a basic course (Grundkurs) and an advanced course (Leistungskurs). Depending on which course has been taken, the weekly hours of mathematics differ from 4 to 6, with some variation between states [7]. From grades 5 to 12, all students in the Gymnasium do 4 h a week of mathematics [8]. At the end of their Gymnasium education, students may sit for the Abitur, essentially the school leaving certificate. The Abitur has two forms, the Allgemeine Hochschulreife and the Fachhochschulreife. The former is the general higher education certificate and will enable students to enter any tertiary institution; the latter is the university of applied sciences entrance qualification and guarantees entry there.2 For example, in Baden-Wu¨rttemberg, the final exams consist of three written ones covering German, mathematics and a first foreign language and one written one on a subject chosen by the student from a second foreign language, biology,

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chemistry or physics, sports, art or music. Not all the subjects that a student studies at this level are examined (available at http://www.stmuk.bayern.de/km/education/ overview/). Generally, students from Realschule take the intermediate school certificate (Mittlere Reife). The natural progression from here is either to start an apprenticeship/training in a medium-level occupation with part-time vocational school or to attend 3 years at a specialized grammar school (in some other states students continue at the Gymnasium). Students from Hauptschule can either go to full-time vocational education or start an apprenticeship. But there is flexibility in the German system as Figure 1 shows. Gymnasium students may leave secondary school at the end of grade 10 and go to a specialized grammar school or even to a vocational school. Similarly, Realschule graduates can go to university via a specialized grammar school and Hauptschule students can go via vocational education to specialized grammar schools.

4. Post-secondary education It is worth noting that in Germany, an emphasis is put on a practical basis to education. In particular, apprenticeships are common (about two-third of all young adults take an apprenticeship [4]) and cover a range of about 350 jobs [3], for which apprenticeships are not the norm in many English-speaking countries. For instance, apprenticeships are common not only for hairdressers, carpenters and motor mechanics (for students largely coming from Hauptschule), but also for master craftsmen, office workers and bankers (for students from Realschule). It should be pointed out that the difficulty of obtaining these apprenticeships varies depending on the economic and other factors. Post-secondary school then, consists of higher education (das Hochschulwesen)3 (containing the two types of university (Universita¨t and Technische Universita¨t), the colleges and the polytechnics (the Fachhochschule)), the specialized grammar schools and vocational education. Vocational education normally takes place in Berufsschule. Students attending these institutions often do so on a part-time basis for the theoretical part of their apprenticeships. For the rest of the time they are employed by a company where they learn the practical aspects of the skills they are developing. Motor mechanics, electricians and carpenters usually go along this route. In 3 years of full-time study, students can graduate from a Berufsschule with an intermediate school certificate and a vocational certificate. They then usually go straight into employment. A range of other schools accepts pupils who have earned an intermediate school certificate. Specialized grammar schools award various certificates after 3 years of full-time attendance (Grades 11–13). These qualify their holders to study at institutions of higher education. The Fachoberschulen offer full-time attendance for at least 1 year or part-time attendance for up to 3 years. After successful completion students are allowed to study at Fachhochschulen [2]. There are 391 institutions of higher education in Germany. These are divided into three groups: the Universita¨t, the Technische Universita¨t (technical universities) and the Fachhochschule. The first two of these are the universities that provide first degrees and graduate work that prepares students for academically based professions. Until recently, the initial degrees undertaken by students in Germany



Figure 2. Numbers of Diplom, bachelor and master degrees in mathematics, 1999–2008.

were the Diplom and the Magister. These degrees took 5 years of full-time study. However, the Bologna Accord (Section 5.1) was signed by Germany in 1999 and the 5-year degrees are to be successively replaced by 3-year bachelor degrees plus 2-year master degrees. From the statistics in Figure 2, we can see that, at least as far as mathematics is concerned, the changeover is proceeding slowly. The Universita¨t offers a range of topics for study from medicine to classics to natural sciences and the arts. The thing that differentiates the Universita¨t from the Technische Universita¨t is the fact that, while the latter also teach a range of subject areas, they concentrate on engineering (and do not generally offer medicine or law). In universities, exams differ from the English models in that the first formal assessment takes place after about half of the Diplom has been completed and oral exams play an important part. This system is changing considerably, however, under the Bologna-induced changes as the new system is much more modular. The Fachhochschulen or universities of applied sciences came into existence in the 1970s and consist of two types, colleges and polytechnics. The latter mainly produce engineers of all types. On the other hand, there are colleges of education, theology, art, music, accounting and management. Although traditionally also awarding the Diplom as a first degree, the Fachhochschule now provides study for bachelor and master degrees. In the polytechnics, 1 year of their Diplom study involves hands-on experience in a company and part of the final assessment is a thesis that looks in depth at an aspect of current professional practice. As a result, graduates from polytechnics are more practically oriented than the ones from the Technische Hochschule. Fachhochschulen do not offer PhDs, but the bachelor and master degrees from both Fachhochschulen and the universities are legally of equal standing. 5. Higher education issues In this section, we look at some of the issues that are currently to the fore in the higher education sector in Germany or are of historical interest. Hence, we consider the Bologna Accord, the Excellence Initiative and current political debate on education. 5.1. The Bologna Accord The Bologna Accord (see, e.g. http://en.wikipedia.org/wiki/Bologna - process# Goals) is a European-wide scheme4 related to tertiary institutions.


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One of the main aims of the Accord appears to be to strengthen the European group of countries as a bloc. This will be achieved by making it possible for students of countries who sign the Accord to study in each other’s countries and for graduates anywhere to have their qualifications readily accepted among the signatory countries. In addition, the bachelor–master pattern of degrees makes Europe more competitive with them for foreign students hoping to study for a first degree. In the past, the cost of a first year degree lasting 5 years may have been prohibitive for many potential foreign students. There has, however, been criticism of the Accord and we list some of these below in no particular order. First, we note that there has been concern over the legitimacy of the process leading to the Accord. It is felt by some that this was driven by the executive branch of government rather than the legislative branch. There is also considerable concern about the reduction of the length of the first degree by 2 years. It is felt that this will lead to a lower skills level and to a less practical and vocational qualification. Some Germans feel that it will not be possible for students to explore their general interests in a 3-year degree. Further, that instead of students receiving an education, they will now be directed into skills that would be of value for them only in the labour market. This leads to a concern by the vocational education sector that universities will take over their role and lead to a decline in the standard of practical skills. It is also debated whether the aim of mobility of students through Europe is likely to be achieved. The claims concerning mobility are thought by some to be excessive. Finally, some people feel that there will be a strong social impact of the reforms. For example, there is a concern that the equality of various social groups and equality between the sexes will be eroded by the increase of fees introduced under the Bologna Accord. It is thought that various groups will have trouble in paying back the loans required for their education. 5.2. The Excellence Initiative The Excellence Initiative was a plan by the Federal government to encourage top university science research with a view to promoting at least one German university into the top echelon of international universities. Two rounds of awards were made to universities in 2006 and 2007 and each award was to be held for 5 years. Annually, E1.9 billion was available, 75% of which came directly from the Federal government. It was to be used in three ways: E40 million were available for the top 40 graduate schools; E195 million were for the creation of clusters of excellence to cooperate with technical colleges, non-university research institutes and economic institutions; and E21 million were for the promotion of research at 10 selected universities. These 10 universities were to be the top universities in Germany and they were to also receive one of the awards for a graduate school as well as hosting one of the centres of excellence. There were criticisms of the initiative. The main ones seem to have been that some of the leading players in the award process had conflicts of interest; that the amount of funds a group already attracted weighted things towards the life sciences and engineering; that this would divide universities into research universities and teaching universities; and that the 5-year lifetime of the awards was too short to allow for sustained development.

International Journal of Mathematical Education in Science and Technology 441 It is worth noting that German universities did rise in the international lists over the period of the Excellence Initiative, but they have fallen back again since. There is still support for a new Excellence Initiative, but it is not clear that the large amount of money required is currently available. On the other hand, consideration is currently being given to a teaching excellence initiative in contrast to the research excellence one.


5.3. The political scene There are at least two factors in the general education debate in Germany. They are: who should control education and how should the system work. Both of these factors apply to education generally, but we concentrate here on the issues with respect to just the tertiary sector. International pressure from the OECD as the result of the PISA and the Bologna Accord has resulted in Germany wishing to increase the number of students who go to university to 40% of the age cohort. This is supported by the Federal government, but they do not control education and there is clearly reluctance on behalf of the state governments to go down that route because of the cost involved. At least two meetings have been held between the various governments without resolving this issue. In Germany, there is also a concern about the university teacher–learner relation. At one point it was common to have members of staff who were employed only to teach. As the result of a conscious policy, most of this group (the so-called Mittelbau) no longer have jobs. Additionally, due to the Bologna Accord, the teaching load will be increased. Therefore, there is a concern among academics that there will be too much teaching and not enough time to do research. The Bologna Accord requires also a restructuring of the curriculum making it more regulated so that there is less freedom in choosing what to teach. The Bologna Accord was intended to simplify and align the university education across the European member states with the goal of allowing students to change seamlessly from one European university to the other. In Germany, this was to be done by breaking the Diplom up into smaller ‘modules’ and so making it easier to learn. A consequence of this system change is a more school-like environment with more regulations and a higher teaching load. Therefore, there is a political debate about the reinstatement of teaching-only positions at universities and even about teaching-only universities. Along with this there is a suggestion that a professor’s teaching load should be increased from 9 to 12 h per week. Many academics and academic organizations, on the other hand, claim that research and teaching are intimately linked and that they should not be separated in the way that is being proposed. They would also like to see a reduction in the amount of administration that they are involved in. They also believe that universities are considerably under funded leading to insufficiently equipped libraries, bad maintenance of buildings and high teacher–student ratios. On the latter point, the overall current professor-to-student ratio is 1 : 58 compared to 1 : 40 some 30 years ago.

6. Enrolment numbers In March 2000, Allyn Jackson wrote an article published in the American Mathematical Society Notices [9] in which he described the disturbing situation


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Figure 3. Total university enrolments in mathematics from 1975 to 2006.

in the enrolments in mathematics departments in German universities. He also discussed the threat to close some departments because of their low enrolments. In this section, we show a more positive set of statistics (supplied by the Statistisches Bundesamt, the same central clearing house from which Jackson obtained his data5). We start by looking at the overall mathematics enrolments in German universities since 1975 (Figure 3). The figures show that Jackson’s article came at the lowest point in mathematics enrolments between 1992 and 2006. Since 2001, except for 2004, student numbers have been increasing and are now at their highest level ever and show a roughly 40% increase in that 6-year period. But absolute numbers can be deceptive. In Figure 4, we show both the overall mathematics numbers relative to the total university student population as well as the proportion of first year mathematics enrolments. These statistics are both positive in the sense that they have been increasing since the end of the last century. First year enrolments are 2.5% of all first year enrolments, while overall mathematics enrolments are about 3% of all university enrolments. So both the absolute and relative numbers of students enrolling in mathematics in Germany are currently increasing. Before we discuss the reasons for the fluctuations in mathematics enrolments shown in Figure 3, we turn to Figure 5. Here, we superimpose the graph of all university enrolments since 1975 on the graph of all mathematics enrolments (with different scales – the all enrolments graph is the one with a more gradual increase and the numbers involved are on the left axis). It is clear that since 1987 the two graphs have approximately run in parallel. From 1987 to 1993 they both increase to a maximum; from 1994 to 1999 they are in decline; from 2000 to 2003 they are increasing; there is a minimum in 2004; and there is a rise from 2004. We note that though the fluctuations appear more pronounced in mathematics than in the overall enrolments, but this may be due, at least in part, to the scaling factor. As a result of the comparison in Figure 5, it would seem that whatever factors have been involved in the changes in university enrolments since 1987, they have affected mathematics equally. (Although as Figure 4 suggests, in recent years mathematics



Figure 4. Relative overall and first year enrolment numbers in mathematics compared to all other enrolments in universities.

Figure 5. A comparisons of all enrolments with mathematics enrolments, absolute figures, 1975–2006.

has slightly increased its market share. It is not clear whether this proportional increase is sufficiently significant for there to be a number of small discernible factors that are affecting mathematics enrolments differently from the overall enrolments.) With this in mind, although mathematicians in German universities at the end of the last century obviously felt under threat, presumably departments in other disciplines were feeling the same pressure. It is worth noting the change in the state of the economy (Figure 6). Apart from 1994 and 1995, the economy increased gradually from 1993 to 2000. There was then a decrease from 2000 to 2003, a rise to 2004 followed by a decline to 2005 and then an increase to 2006.


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Figure 6. State of the German economy, 1992–2007.

Mathematics enrolments appear to be going the opposite way. They declined from 1993 to 1999; increased from 2000 to 2003; declined for 1 year to 2004; and then have increased since. Since 1993 it would seem that many potential students might have sought employment rather than enrolling in a mathematics degree at a time when the economy was buoyant and when jobs were available. On the other hand, they enrolled in mathematics when the economy was on the decline. Actually, from Figure 5, it would seem that the real period of concern for mathematicians should have been the years up to 1987 when mathematics was showing a decline in numbers while the overall student numbers were increasing. At this point, we need to mention that the mathematics enrolment numbers consisted of two types. First, there were students who were going directly towards the Diplom; second, those who were undertaking training as secondary mathematics teachers. In fact, at the start of the decline, teacher trainee numbers were significantly higher than Diplom enrolments. Over the period of the decline, the enrolments in teacher numbers were also declining because of an over supply of teachers. On the other hand, the Diplom mathematics enrolments were increasing for the whole period just as the overall enrolments were. The decline in mathematics enrolments, therefore, appears to be attributable entirely to the downturn in teaching positions. (It is interesting to compare this with the teaching figures in France, see Arnoux, et al. [10].) If we now consider the data regarding the number of school leavers who have qualified for university since reunification, it shows, not surprisingly, that there are more qualified students than the ones who actually go to university. The number of school leavers, though, shows a gradual rise over the period and there are years when significant numbers of student do not take up the option to attend university (this may well be due to the economy). First year mathematics enrolments, though, follow a similar curve to both overall first year enrolments and science enrolments and so there is no discrimination against mathematics as a subject as far as we can see. We now turn to gender issues. There are two aspects here that we will highlight; the first is to absolute numbers and the second is related to inferred graduation rates.



Figure 7. Enrolments of all male and female students at universities and universities of applied sciences.

There are certainly more males than females in higher education in Germany but female numbers (in all subjects) became greater than male numbers in universities in 2001 (Figure 7). It is interesting to note, though, that in Fachhochschule the number of males is significantly greater than the number of females (by more than 100,000). The difference in both cases appears to be getting larger. The explanation for the Fachhochschule may well be that these are divided into two types, one preparing future primary school teachers and the other largely training engineers of some description. The latter group has a bigger enrolment and the jobs for which they prepare students have been traditionally male ones. The situation for first year students is only slightly different. Female first year enrolments overtook male enrolments in universities in 1995. There seems to be no convergence for Fachhochschule. At first sight it is a little surprising that overall female enrolment numbers in universities took until 2001 before being larger than the male numbers. Similarly, if we look at the gender balance in mathematics at universities, we see that first year female students have also outnumbered their male peers since 1995 but, overall, male numbers still exceed female numbers in 2006 though the margin is small6. The reason for the surprising delay in women not yet overtaking men in overall mathematics enrolments can be found by looking at the cumulative graphs in Figure 7 and again it involves the secondary teaching factor. First, we recall that the Diplom is a 5-year degree and that this is being phased out and replaced by the bachelor/master degrees. However, pre-Bologna, if we add the first year intakes for 5 successive years we should get a reasonable approximation to the total number of students in a given course. In Figure 8, we did this for the females in mathematics and the cumulative estimates approximated the actual mathematics enrolments quite well. You would expect this to provide an underestimate as some students enter the course after their first year of enrolment, and some students delay taking their final examinations. On the other hand, this may be balanced by students leaving the course. Anyway, pre-Bologna, the 5-year


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Figure 8. Cumulative numbers of first year students for females over 5 years and for males over 7 years.

cumulative estimate of female mathematics student numbers is a good approximation to overall female mathematics enrolments. In Figure 8, we also repeated the cumulative exercise for males. In contrast to the female graph it took an accumulation of 7 years before we had a reasonable approximation of the overall male mathematics student numbers. This seems to indicate that men take 2 more years than women to complete their Diplom7. However, on looking deeper into the data, the reason for the 2-year lag seems to be that a larger percentage of the women in Figure 8 are studying to be secondary teachers than men. In their degree they do less mathematics than if they were taking the Diplom and they also take non-mathematics subjects as part of their degree. Hence, their course is not so difficult as the Diplom. Men, of course, study to be secondary teachers too but not in such large numbers as women. Consequently, it does not affect the men’s graph in Figure 8 as it does the women’s graph. But this does not totally explain the situation. When we separate out the secondary teacher and Diplom enrollees and use our cumulative approach to approximate overall enrolments, it seems that women take 5 years to complete their teaching courses and nearer 6 to complete the Diplom. In comparison, men take 6 years to complete their teacher training and 7 to complete their Diplom. As a result, we can confirm that it is easier to get a teaching degree than the Diplom, but we can also confirm that men take longer to get either qualification compared to women. One reason for this may come from the survey that was taken for the Year of Mathematics 2008 (available at http://www.jahr-der-mathematik.de/)]. It appears that only about 50% of students are concerned to finish their Diplom in minimum time. More students (around 75%) want to make sure that they obtain the best results they can. So it may be that men are delaying their completion deliberately in order to get the best degree possible and, presumably, make them better prepared for the next step in their career. There are two other points worth making here. First, in both of the graphs of Figure 8, the cumulative sums exceed the actual enrolment numbers in 2003/2004.


International Journal of Mathematical Education in Science and Technology 447 There is clearly some reason why all students are suddenly completing their degrees more quickly. This reason is the introduction of credit points. Previously, assessments occurred throughout the Diplom without credit and the final degree was awarded solely by a final examination. With the introduction of credit points, all assessments were counted towards the Diplom and as a result there was less reason to put off the final examinations. This credit point approach to final assessment is along the lines adopted in the new degree structure after the Bologna Accord. In passing it should be noted that the fact that cumulative totals exceeded actual enrolments in 2003/2004 was not because of the introduction of the Bologna Accord as might have been expected. It seems that the numbers of students enrolled for bachelor/master degrees is not large (less than 20% of overall enrolments in mathematics; Figure 2) and so has only a small impact on statistics as shown in Figure 8. The second point worth making is that traditionally German universities expect a low completion rate. The data shows that less than 50% of all first year students who enrol in mathematics graduate in minimum time. Apparently, mathematics departments expect to only have ‘the best’ students in the final year of the Diplom. (Actually, some students transfer from other subjects into mathematics. Their chances of success are even lower than those who enrol directly into mathematics from school.) This clearly differs from other countries [11] where academics work to ensure that as many students graduate as possible. However, it should be noted that first year mathematics students are more optimistic about their future careers than first year students in all other fields of study (86% versus 76%, [12]).

7. Discussion It is worth summarizing the results of the data that we have collected. From what we have seen above, it first seems clear that mathematics is not disadvantaged with respect to other subjects in Germany. Unlike France, there seems to be no movement away from mathematics in universities even though a good percentage of the students who enrol in first year will not complete their Diplom. (It should be noted that the same phenomena is present in other subject areas.) Second, when absolute numbers in mathematics were declining at the end of the twentieth century (available at http://www.jahr-der-mathematik.de/), the numbers were declining in other subject areas. A more pronounced decline with respect to other subjects had occurred earlier than that because of the decrease in secondary mathematics teachers, but this cause of change seems to have disappeared. Third, mathematics enrolments in German universities seem to be inversely related to the state of the economy. When the economy is going well, fewer students appear to enroll in the subject than at other times. Consequently student enrolments in Germany, as elsewhere, appear to be more related to the general state of the job market than to the subject being studied [1]. This is certainly consistent with the data that exists from other countries, particularly France [10]. However, it is not consistent with China [1] where, as the economy seems to be booming, many more students are enrolling in mathematics. This difference between Germany and China may be due to the fact that China is currently developing rapidly. Finally, there was a period in which women appeared to take roughly a year less on average to complete their Diplom than men. But this may be because more women

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were taking a secondary teaching degree in which the mathematics is less deep, while male students are taking extra time to complete their Diplom in order to obtain a better result.


Notes 1. It should be noted that most of the references to Federal or state bodies in this article are in German. In the references, we have translated the titles of those documents. Ref. [2], however, is available in English. 2. The Fachhochschulreife can also be taken at certain specialized grammar schools. The difference between grammar schools and specialized grammar schools is that specialized ones are oriented towards occupations [2] and the Gymnasien towards academic pursuits. 3. Hochschule is often wrongly translated into English as ‘high school’. 4. Though other countries are becoming involved in some ways. New Zealand, for example, is introducing a diploma supplement that will give more details of what a graduate actually studied for their degree. 5. Unlike many other countries, Germany has good data that is readily available. 6. Incidentally the number of males in science still easily exceeds the numbers of females. 7. The 6-year cumulative totals are a better approximation for males up to about 1990, but even this still means that in 1981, say, men took longer to complete their Diplom than women.

References [1] D.A. Holton (ed.), Special issue of iJMEST, 14 (2009). [2] Federal Ministry of Education and Research, Education in Germany – Basic structure of the education system of the Federal Republic of Germany. Available at http:// www.bmbf.de/pub/bildung_in_deutschland.pdf. See also the Bayern model at http:// www.stmuk.bayern.de/km/education/overview [3] Portal fu¨r Grund-und Strukturdaten 2007/2008, Bundesministerium fu¨r Bildung und Forschung. Basic data 2007/08. Ministry of Education and Research. Available at http:// gus.his.de/gus/einfuehrung0.html [4] Bildungspla¨ne der Bundesla¨nder fu¨r allgemeinbildende Schulen, Principles of curricula and courses of instruction for general education of all Federal States of Germany. Available at http://www.bildungsserver.de/zeigen.html?seite¼400 [5] Hauptschule bis zur Jahrgangsstufe 10 anhand der Beispiele der Bundesla¨nder BadenWu¨rttemberg, Bayern und Nordrhein-Westfalen, Secondary general school education till grade 10. Available at http://www.schule-bw.de/schularten/hauptschule/; http://www. schulberatung.bayern.de/schulberatung/bayern/schullaufbahnberatung/schullaufbahnen/ hauptschule/index_05631.asp; and http://www.callnrw.de/php/lettershop/download/816/ Druckfassung%202005.pdf [6] Statistisches Bundesamt, 2008. Available at: www.gus.de/gus [7] Handbuch Abitur – Leitfaden fu¨r die gymnasiale Oberstufe, 52 pp. Available at http:// www.schule-bw.de/schularten/gymnasium/zentralepruefungen/abitur/kurswahl [8] Stundentafeln und Kontingentstundentafel fu¨r allgemeinbildende Schulen in BadenWu¨rttemberg, Curriculum at Grammar Schools in the Federal State of BadenWu¨rttemberg. Available at http://www.gbg.wn.schule-bw.de/strutafl.htm [9] A. Jackson, Declining student numbers worry German mathematics departments, Amer. Math. Soc. Notices. 47 (2000), pp. 364–368.



[10] P. Arnoux, D. Duverney, and D. Holton, The rise and fall of mathematical enrolments in the French educational system: a case study, Int. J. Math. Ed. Sci. Tech. 40 (2009), pp. 43–57. [11] J. Oikkonnen, Ideas and results in teaching beginning math students, Int. J. Math. Ed. Sci. Tech. 40 (2009), pp. 127–138. [12] K. Briedis, T. Egorova, U. Heublein, M. Lo¨rz, E. Middendorff, H. Quast, H. Spangenberg, Studienaufnahme, Studium und Berufsverbleib von Mathematikern. Einige Grunddaten zum Jahr der Mathematik. (Studying Mathematics and Careers for Mathematicians. Some basic data for the Einstein Year), September 2008, 98 pp. Available at http://www.his.de/publikation/forum/index_html?reihe_nr¼F9/2008