ABSTRACT

The anniversaries of Felix Klein (1849–1925) in the years 2024 and 2025 are going to give rise to appreciations of his influence on the mathematics education in the German, in particular Prussian, “Gymnasia”. The usual narrative is that the Meran reform that he had initiated brought at least infinitesimal calculus, if not the notion of function at all, to this type of secondary school. (The name “Meran reform” goes back to the city in which the annual meeting of the German mathematical society took place in 1905 that approved the plans for the school curriculum drawn up by Klein and his collaborators.)

 

Evidence for this view of Klein’s role is often taken from comparing the official school administration documents (“Lehrpläne”), in particular the first document on the final examination of the “Gymnasium” from 1788 and the collective description of the curricula for secondary schools from 1882. However, little is usually said about the time in between, which lasted for almost a century.

 

A closer study shows that the story was not that simple and interesting things happened during this time. For example, in 1829 the Prussian authorities issued an order that banned infinitesimal calculus from the “Gymnasia” since it was considered too difficult for the pupils – which, in turn, implies that it had been topic of the education at least at some schools before. In addition, many school teachers did not want to do without the exercises for determining local extrema and therefore tried to get around this ban. The most prominent example was the Berlin mathematics educator Karl Heinrich Schellbach (1804–1892) who published a method inspired by the ideas of Pierre de Fermat (1607–1665) that avoided the open use of infinitesimal ideas. Schellbach’s writings also show that the term “function” was standard for Prussian mathematics teachers in the mid-19th century. Furthermore, even after the Meran reform of 1905 mathematics educators like Max Simon (1844–1918) still insisted on a treatment of functions without the use of infinitesimal ideas.

 

SELECTED BIBLIOGRAPHY

Burscheid, Hans-Joachim (1984). Eine Schulenbildung unter den Gymnasialdidaktikern des ausgehenden 19. Jahrhunderts, Zentralblatt für Didaktik der Mathematik, 16, pp. 191–195.

Mattheis, Martin (2020). Wie der Funktionsbegriff in die Schule kam. Siegener Beiträge zur Geschichte und Philosophie der Mathematik, 13, pp. 155–173.

Schellbach, Karl Heinrich (1860). Mathematische Lehrstunden, Aufgaben aus der Lehre vom Größten und Kleinsten, A. Bode & E. Fischer (Eds.), Berlin: G. Reimer.

Schubring, Gert (2022). The History of ICMI: The First Phase as IMUK and CIEM. In F. Furinghetti & L. Giacardi (Eds.), The International Commission on Mathematical Instruction, 1908–2008: People, Events, and Challenges in Mathematics Education, Chapter 1. Cham: Springer Nature.

Ullrich, Peter (2021). Karl Schellbach (1804–1892) und seine Beiträge zu Mathematik, Lehrerbildung und Wissenschaftspolitik. In H Fischer, T. Sauer & Y. Weiss (Eds.), Exkursionen in die Geschichte der Mathematik und ihres Unterrichts (pp. 223–234), Münster: WTM Verlag.