Mathematics degree

from Wikipedia, the free encyclopedia
Math lecture

There are three types of degree programs in mathematics in German-speaking countries: the classical diploma , the classical teaching degree and the new bachelor and master’s degree programs . In addition, mathematics can sometimes be taken as a minor . There are also numerous specialized mathematics courses such as industrial mathematics , business mathematics , financial mathematics or biomathematics . The subject is offered by universities and technical colleges as well as by technical colleges.

There were no admission restrictions for mathematics courses at most universities in the current decade. General hurdles faced by universities for admission to further study courses and qualifications ( master’s , doctorate ) are possible.

Diploma course

The classic diploma course is divided into a four-semester basic course and a five-semester main course. The actual study times can differ significantly from this standard study time.

Undergraduate studies

The basic course is quite uniform. It usually consists of:

The contents of linear algebra and analytical geometry are algebraic structures , vector and matrix calculus , in particular the Jordanian normal form , affine spaces and their geometry, projective geometry , symmetry groups , isometries in n-dimensional real and complex spaces and module theory . The focus is on proof techniques and constructions.

The Analysis I deals with the axiomatic construction of the differential - and integral calculus in real and complex one-dimensional spaces, and the foundations of topology . The Analysis II considered the multidimensional differential calculus and ordinary differential equations . In the further course of the analysis course, multi-dimensional integration, measurement and integration theory, theory of manifolds , vector analysis , Fourier analysis , introductions to differential geometry , function theory and functional analysis are dealt with. The detail, scope and focus are very much dependent on the university and usually also on the lecturer.

Applied mathematics consists of one or two lectures on numerics, alternatively or in addition to a lecture on probability theory and statistics and, in some cases, an introduction to computer science .

The minor subject usually has a somewhat smaller scope than the basic course of the corresponding subject in the teaching profession for the upper secondary level.

Main course

Formally, the main course always consists of the areas of Mathematics I, II and III as well as the minor. However, the design differs depending on the university.

Traditionally there is the division

and the division

In each of the three areas, two to three consecutive lectures and associated seminars must be attended. Often, however, there are conditional dependencies that raise the actually necessary study effort far above the formal requirements of the study regulations . In order to understand higher differential geometry, it is usually necessary to acquire extensive knowledge in the areas of topology, algebraic topology , differential topology , differential equation theory and complex analysis , whereby differential equation theory cannot be understood without a deeper study of functional analysis, etc.

Pure math

In pure mathematics, all areas to choose from, which is under the generic names algebra , number theory , topology , group theory , logic , Algebraic Geometry , Algebraic topology , differential geometry , differential topology , differential equations , partial differential equations , Lie groups , functional analysis , measure and integration theory , Representation theory , complex analysis , cohomology groups etc. can be summarized. Since there are strong interdependencies between these areas, pure mathematics must be studied very extensively, which is not taken into account in the study regulations for formal reasons (standard study times, etc.).

Applied Mathematics

Applied mathematics includes the areas of probability theory , statistics, stochastics and numerical mathematics, often also mathematical physics and biomathematics . Statistics accesses probability theory, which is often combined with it in a course series. To understand probability theory, measure and integration theory are required. Numerical mathematics often falls back on basic knowledge of functional analysis.

Thesis

Diploma theses in mathematics officially take six months, sometimes less at universities of applied sciences. The actual duration can be considerably longer in some faculties, so the official duration is simply the time to compile the results.

Minor subjects

In principle, all subjects from the offer of the respective university that are meaningfully related to mathematics can be selected as minor subjects. In some courses, e.g. B. Financial Mathematics, the minor is already established. The scope of the minor in the basic course corresponds roughly to the scope of the subject as one of 2 subjects in the teaching degree for grammar schools or is slightly less, in the main course it corresponds roughly to a master's minor .

Typical minor subjects are:

but often also:

Teacher training

In principle, the teaching degree is regulated by state law. The courses of study differ considerably from one another. In Baden-Württemberg, for example, prospective teachers, with the exception of high school teachers, are trained at special pedagogical universities . For the grammar school teaching post, however, common structures can be identified nationwide.

teacher in grammar schools

Undergraduate studies

The basic course for teaching in mathematics - with the exception of teaching at the primary level - generally corresponds to the basic course for the diploma course. In addition, achievements in the second subject and in educational science have to be completed. Most mathematics faculties do not take this into account, even if the certificate requirements are sometimes slightly lower than in the diploma course. The academic advisory service often recommends studying mathematics at first and neglecting the second subject and education. However, this can have negative consequences for funding for BAföG recipients or scholarship holders.

Main course

The main study of the grammar school or secondary level II teaching post in mathematics differs not only from state to state, but also from university to university. As a common framework, it can be seen that there are certain areas that are always included - albeit in different forms. An example of a breakdown that can be found at universities in North Rhine-Westphalia:

  • Algebra and Number Theory, Group Theory (A)
  • Geometry and Topology (B)
  • Analysis, application-oriented analysis (C)
  • Statistics and Stochastics (D)
  • Mathematics Didactics (E)

Algebraic geometry and algebraic topology can be assigned to both (A) and (B), differential geometry and differential topology can be assigned to (B) and (C). Application-oriented analysis means, for example, numerics.

The scope and quality of the didactic training is very dependent on the location.

Model projects for a Bachelor-Master model in teacher training are also running at Bielefeld University and the Ruhr University Bochum .

Other teaching posts

The other teaching posts are regulated very differently, as each federal state has a different structure of the various teaching posts. Depending on the country, there are two to five different teaching posts. There is still a certain degree of comparability in the area of ​​primary / primary school. But even there, the course can hardly be represented schematically because the differences between the countries are too great.

Teaching qualification for primary level / primary school

The primary school teaching position differs from other teaching positions u. a. because in many federal states it is compulsory to include mathematics or language education (German). In mathematics, special lectures on topics such as arithmetic are provided in some federal states ; in other states, the primary school teaching post is regulated jointly with that of the secondary level 1 teaching post. Occasionally, however, the training can be quite different within such a joint teaching post.

Information on this is available from universities and education ministries in the federal states. Since even the type of university (university, pedagogical university of applied sciences) is different, it is important to consider before starting such a degree that the teaching qualification is only acquired for the specific federal state in which the university is located. However, this applies a priori to all teaching posts.

Teaching positions for secondary level 1 / secondary schools / secondary schools / regular schools

Here, the variety of school types is already so great and differs from state to state so much that it is practically impossible to make any cross-border statements.

Bachelor and master studies

The bachelor's degree takes six semesters, the master's four semesters. The first three to four semesters mostly correspond to the previous basic course in the diploma course, after that there are different models.

In Bielefeld and Bochum there are also models that design teacher training consecutively .

Bachelor degree

The bachelor's degree lasts six semesters. Its content differs significantly from university to university. However, there is a tendency: The first two semesters are included in the classic or another form

  • Analysis I
  • Analysis II
  • Linear algebra I.
  • Linear Algebra and Analytical Geometry II
  • partly also introduction to computer science / programming

In addition, are mostly

  • Analysis III or vector analysis
  • Function theory I
  • Numerics I and / or stochastics

intended. A number of other events are usually included in the curriculum, but these are completely different from Bachelor to Bachelor. This list is by no means binding, but only a list that applies to most courses of this type.

In most cases, there is also a minor that is similar to the one in the diploma. Typical minor subjects are again physics, economics, computer science, and electrical engineering. At some universities, minor subjects with more limited content are offered; For example, at the Ludwig Maximilians University in Munich , the minor subjects of computer science, statistics, experimental physics, theoretical physics, economics, business administration, insurance and risk management are offered.

Bachelor in high school teacher training

For the consecutive teacher training course, these contents are also binding, with the exception of function theory I (the basics of function theory are then included in the analysis curriculum) and numerics I (instead of these, however, related services including an introduction to programming are required), as well as one Introductory course in the fields of statistics, stochastics and probability theory. The further scope of studies corresponds roughly to the classic teacher training course (however with deviating area specifications), although in the bachelor's degree only subject-theoretical components are provided, the didactics only appear in the master's ( Bochum model ), function theory and numerics can of course be incorporated into the areas.

The recognition of academic achievements from such models is only regulated in the relevant federal state and also there with reservations, such as time limits. Thorough information about this is urgently recommended before starting such a degree.

Master's degree

The science and business-oriented master’s program no longer follows more general rules.

Master in high school teaching

The model experiments for graded teacher training are i. d. R. (Land law) Deviations from the teaching qualification examination regulations (LPO) of the respective country are allowed. Nevertheless, there is a certain orientation towards content. There are 2 models in North Rhine-Westphalia:

  • The Bochum model : The theoretical part of the course (everything attributable to mathematics except for didactics) is almost entirely part of the bachelor's degree, didactics, as well as educational sciences and a certificate of achievement from the remaining mathematics as well as the didactics of the other subject and a further certificate of achievement from the other Subject are accommodated in the master’s.

In addition to studying both subjects, the Ruhr-Universität Bochum model also provides for what is known as an optional area, in which, under certain conditions, interdisciplinary events, foreign languages, computer and rhetoric courses, internships, etc. can be introduced. Of course, school-related services, such as school internships, can be included within this area, but they do not have to. However, this is in contrast to the decisions of the Standing Conference of the Ministers of Education and Cultural Affairs on model experiments for tiered teacher training, which makes permanent recognition or recognition outside of North Rhine-Westphalia rather unlikely.

  • The Bielefeld model: One subject and educational sciences or one subject in full and one further subject are allocated to the Bachelor, the rest to the Master.

Time spent studying

Formally, the time required for (most) lectures in the undergraduate / bachelor’s degree is four hours of 45 minutes each. (Almost) every lecture includes a two-hour exercise. Seminars are two to three hours long , block events are correspondingly longer. In the main course / master’s degree, the lectures are two to four hours long (depending on the university, lecturer and degree), for which two-hour exercises are often offered.

Additional time is required for solving exercises, depending on the one hand on your own abilities and on the other hand on the level of difficulty. The lecturer or one of his employees creates the exercise. Depending on the lecturer, the time required to solve the exercises in a lecture can vary greatly. The formally specified effort for an event ( e.g. through ECTS points) can differ greatly from the actual effort.

promotion

A doctorate can be taken at universities and technical colleges that offer this subject. Up to now, doctorates have been available as so-called external doctorates , i.e. H. possible without enrollment for doctoral studies, as well as doctoral studies . Individual doctoral regulations can differ from this. Changes could be made here in the course of the Bologna Process .

The doctorate concludes with a doctoral degree , which can include the scientific doctoral degree Dr. rer. nat. as well as the philosophical degree Dr. phil. This depends on the respective doctoral degree regulations. In the course of internationalization, the Anglo-American Doctor of Philosophy (philosophiae doctor) Ph.D. awarded.

So far, doctoral degrees in mathematics have varied widely. They start with a few months - then it is mostly the continuation of a diploma thesis lasting several years - up to many years (as a rule, due to the limited employment opportunities for qualification positions after 6 years, continuation is no longer possible due to insufficient funding.).

labour market

Mathematics is a degree at the highest level of abstraction, but mathematicians have comparably good prospects, as the possible applications have multiplied in the last few decades:

  • The software industry asks mathematicians very often; their area of ​​application is similar to that of computer scientists .
  • At universities and research institutions, there is increasing demand for mathematics graduates again , but this remains a very manageable job market.
  • Schools increasingly need specialist mathematics teachers, and a greater shortage of mathematics teachers is also expected in the grammar school teaching post (cf. teacher training course ).
  • Banks and insurance companies mainly use mathematicians in the IT sector and as financial mathematicians (e.g. for analyzing exchange rate risks) or actuaries .
  • Industrial companies employ mathematicians - in addition to the field of computer science - wherever complex problems need to be solved (e.g. in development departments in the aerospace industry ).
  • The public service also recruits mathematicians, particularly in statistical offices and authorities (e.g. the Deutsche Bundesbank or the ECB ).
  • In isolated cases, auditing firms or engineering firms are also looking for mathematicians.
  • Management consultancies value the thinking and analytical skills of mathematicians and employ them as well.
  • Other typical employers for mathematicians are secret and intelligence services .

On the (free) job market, mathematicians are always in competition with other disciplines - to varying degrees depending on the industry or specialization requirements.

Related fields of study

Technomathematics / engineering mathematics

The study of technical or engineering mathematics is very similar to the classic diploma course. It is practically a diploma course in mathematics with an engineering minor, but the depth exceeds the usual minor. Further lectures and exercises in computer science are also planned.

Technomathematics is offered by some technical colleges and universities. The first bachelor programs already exist. Universities of applied sciences also offer courses in technical mathematics.

Business mathematics / financial mathematics

Business mathematics, financial mathematics or even more specialized actuarial mathematics are courses of study that can be found more frequently at universities, but also occasionally at universities of applied sciences.

It is a course of study that is similar to the classic mathematics course with a minor in economics, but includes more applied mathematics and computer science. In addition, specific courses on financial and actuarial mathematics (specialty of applied mathematics) are offered.

Applied Mathematics

Various courses of study can be found under the general name Applied Mathematics

  • Ordinary mathematics courses (diploma, bachelor's degree) with a reduced proportion of pure mathematics and an increased proportion of applied mathematics and computer science
  • the business mathematics course
  • the Technomathematics course

correspond.

statistics

There are independent statistics courses in Germany at two universities: at the Technical University of Dortmund as a diploma course, since the winter semester 2007/08 only as a Bachelor / Master course, and at the Ludwig Maximilians University in Munich. Another bachelor's degree in "Applied Statistics" exists in Magdeburg, which is supported by the Otto von Guericke University Magdeburg and the Magdeburg-Stendal University of Applied Sciences. A purely university master's degree in statistics is offered at the university there. In Switzerland, the ETH and the University of Neuchâtel offer statistics master’s courses in English.

The statistics course in the first 2 semesters is very similar to the diploma or bachelor's / master's degree in mathematics:

  • Linear Algebra I and II, but with application-oriented exercises in terms of statistics
  • Analysis I and II, also with reference to statistics in the exercises

Furthermore:

  • Introduction to Statistics
  • Data processing
  • Events on statistical modeling

In the higher semesters, the course specializes more and more on statistical application, although the very strong theoretical orientation that is usual for university courses is retained.

For more information, see Statistics (subject) .

See also

Conference of the German-speaking mathematics faculties

Web links