Maybe I should write this in German. But why not in English? After all, we need more European standards in education. And the discussion may be of interest to students in other nations. Be aware, however, that the levels of mathematics in schools are quite different between countries like India, China, Germany or the USA. But all have something in common. And that will be the point of this posting.
Currently, we have an uproar about the 2019 mathematical test for the German Abitur. In case you are not familiar with the German system, the Abitur is a final examination of the highest level of school education in Germany. It is usually written at the age of 17/18 after 12/13 years of school life and qualifies for an entrance to the Universities. The test contains centralized elements now for all of Germany to guarantee the same level for all states in Germany (the Bundesländer). By law, each state is responsible for education in its schools. The teachers have to select sets of problems from this pool to make the results more comparable.
The students claim that the test was much more difficult than last year, and contained unexpected problems. The final word about this will be out soon when the grades of this year and last year can be compared. However, petitions were signed by thousands of students to lower the requirements this year. Obviously, not only participants in the test signed. This shows the public interest in the matter, and also the problems with mathematical education in the general public.
A typical example of this attitude I heard in a radio interview with a mathematician who organizes the Mathematical Olympiad in Germany. The question was: „If the math test in the Abitur was already that difficult why do we need mathematical Olympiades?“ It simply reflects the general tenor of „Who needs mathematics?“. The interviewed mathematician replied very politely, where I would have been much more direct and would have asked: „Why do we need a high jump?“ The interview goes on with asking if the math problems of the Abitur could be used for the Olympiade. Again, the answer was polite, but a bit off in my opinion: The Abitur is more on reproducing known techniques, and the Olympiade is on creating ingenious solutions. This answer made me write this posting. I think it is wrong and misleads the public view on mathematics.
The truth is that the successful participants of the math Olympiad all practice quite a lot. They have meetings, they read books, and they study old problems. I once hosted one of their sessions and saw how serious they take mathematics. They have a lot of tricks up their sleeves that the innocent student has never heard of. Practicing is the secret to success. It is the same as in all other competitive activity, including the Olympic Games. You do not expect a high jump professional to be excellent in table tennis. It is not what he learned and improved with thousands of repetitions. And being an expert in Algebra does not make you a brilliant applied mathematician, as well as the other way around. Human intelligence relies on learning and recognizing, and not on brilliant extraordinary ideas. Those happen only to the ones that have studied the most.
Getting back to our „Mathe-Abitur“, we have to face the truth that we have to teach the necessary skills before we can ask students to apply them. There is no principle difference between reproductive tasks and so-called transfer tasks. It is just so that the easy problems make it obvious which skills have to be applied, and the difficult problems hide this and may require more than one skill. The student needs to learn to sort out the mathematical content from the given text and see how he can apply the learned material to the problem. This is a skill of its own.
In summary, we cannot really judge the difficulty of an exam without knowing the details of the teaching that prepared for the exam. The only other way is to study previous exams. Clever students do, of course, look at these on their own. For most students, this must be part of the teaching. At this time, the only way to convince me that the problems were too hard would be a petition by the math teachers. They are the experts. A serious drop in the grades would also need an investigation, with input by those that really face the students, the math teachers.