Considerations on the development of algebra in teaching. (Q1181037)

The author asks the very interesting question: Which time is passing until new results become subject in lectures. The ideal case would be: A mathematician gets new results which are distributed in talks and published in a journal. Other specialists pick them up and integrate them in their lectures. Textbooks including or based on the new results are the next step. As a consequence even non specialists present the new results in their lectures. But reality is different, this is proved by a case study concerning algebra in the first half of this century at the university of Würzburg. This is only possible because the left papers of Otto Volk include lecture manuscripts not only of his own but also of his celleagues; but there were also used some manuscripts contributed by others. Heinrich Weber's textbook ``Lehrbuch der Algebra'' (1895) became a classical for the next 3 decades. Though algebraical structures like group and division ring were not excluded, the author focussed on the theory of equations culminating in Galois theory. Weber's textbook was the source of Georg Rost whose special field was not algebra but function theory and astronomy. During his algebra lecture in 1904/5 Rost relied basically on Weber, dropping Galois theory. Like Rost Emil Hilb was not a specialist on algebra, his algebra lecture of 1927 was a classical one. At the same time new textbooks were published by Oskar Perron (1927), Helmut Hasse (1926/7) and Otto Haupt (1929). All these authors adopted a more modern view concentrating more and more on structural concepts than on equation theory; especially Hasse was mainly influenced by the new results of Toeplitz and Steinitz. Though Otto Haupt was at first a student of Hilb, he afterwards followed the concepts of Emmy Noether. Hilb still in contact with Haupt now revolutionized his algebra lecture from 1928/29, unfortunately he died in the same year. Otto Haupt's textbook did not manage the break-through, this was due to B. L. van der Waerden, his ``Algebra'' of 1930 became the one textbook for the next generations. Though Otto Volk was familiar with the new concepts, his lectures were based on history. In his algebra lecture of 1938/9 he started with the Greek and did not reach modern times. The break-through for modern algebra was achieved by Hermann Schmidt, though his main field was analysis. Schmidt was professor in Würzburg from 1951-1970. In his algebra lectures he began with fundamental concepts as sets, lattices, quasiorders, structures of extensions, polynomial rings etc. It was only under Schmidt that structural algebra became the standard knowledge of future teachers. But at the same time teachers opposed to that modern view. They thought that modern algebra was of no use for teaching algebra in high-schools. The didactics of mathematics would be able to build a bridge between the mathematics of universities and of schools.