General Advice on Beginning a Course of Study in Mathematics

Working in mathematics can be fun, even as a beginner: Whether you examine a series for convergence, calculate an integral (in your head, not by pressing a button ;)), find a small proof or a counterexample to a conjecture yourself, or solve a system of linear equations or a differential equation - if the task was sufficiently difficult for you and you were able to solve it on your own, you will experience joy at this kind of success.

Don't let anything or anyone spoil this fun! This way, studying mathematics will help you - almost regardless of the content - gain a key methodological qualification, on the basis of which you will be offered excellent career opportunities both in classical occupational fields and in typical new occupational fields at the present time.

Beginners' mathematics lectures should actually be an exciting experience for students through giving insights into the development of mathematics since Pythagoras and Euclid, over Leibniz and Newton to modern questions and today's research, and through highlighting the role of mathematics in intellectual history and its importance for us today. Sometimes they succeed at least partially in achieving this ideal teaching goal.

Often, however, the difficulties in detail outweigh for beginners. If you have taken an advanced course in mathematics, you will find some of the school material here again in a more in-depth form; this will make it easier for you to overcome the initial difficulties. Some of you who have enjoyed particularly good teaching at school may even get the initial impression of not being offered anything new. In that case, don't become impatient and miss the moment, which will probably come soon, when the lecture material goes beyond your previous knowledge.

Many times it will also be the case that you have already heard of certain mathematical terms and facts at school, but that they are covered in more detail and more thoroughly here. Mathematics, as you will encounter it here at the university, can only be compared to school mathematics to a limited extent. It is characterized by:

• the axiomatic structure,
• the precise mathematical language and
• the high level of abstraction.

The axiomatic-deductive structure and the use of abstract conceptualizations are not indispensable in the basic lectures, but they have become common today. This way of thinking has arisen in the solution of the great classical problems of mathematics, and it has proved itself in the endeavor to make the developed theories valid and applicable in as large a field as possible. In school, a consistent axiomatic presentation of mathematics is not possible for didactic reasons. Therefore, at university, a longer period of familiarization is usually necessary for you to get used to it. At this point, it should not go unmentioned that there are also dangers if you are confronted exclusively with ready-made theories in an axiomatic-deductive structure in the early semesters. The danger is that you do not recognize the original problems from which these theories have developed and that you misunderstand the theory presented to you as an end in itself and as the actual content of mathematical research. Unfortunately, such misunderstanding is not uncommon today and is even beginning to have an impact on curricula and textbooks. Do not be impressed by this; always look for the motivation and historical background of the theory offered to you in the lecture and ask the instructors about it as well. An elegant theory is always only an end state in a development prompted by definite problems.

For some, the transition from the classroom to the introductory mathematics lecture can come as a shock that is difficult to overcome. School teaching is often still geared toward computational techniques and procedures. At university, however, the focus shifts towards the analysis of abstract structures and principles. This requires readjustment. Lack of prior knowledge from school may be partly responsible for adjustment difficulties. While our basic lectures are generally structured in such a way that no prior knowledge is required in a formal sense, the assumption is, however, that you are sufficiently familiar with mathematical objects and methods taught in school. This is reflected in the manner of presentation and in the speed of the procedure. For this reason, good knowledge of school mathematics and skills acquired in school are very useful.

There is no one-fits-all solution for overcoming the initial difficulties. In any case, it can be said that the successful completion of a study program in mathematics requires (a certain specific) talent, genuine interest, and the willingness to work persistently and with concentration. If at least one of these requirements is not met, studying mathematics will sooner or later become a difficult burden or disappointment to bear.

At this point, we would once again like to point out the importance of attending the mathematics pre-course offered by the student council.