## The reform of mathematics teaching in Hungary

We quote below an extract from L Kovács, Teacher László Rátz, in

*Rátz László tanárúr*(Berzsenyi Dániel Föiskola, Szombathely, 2006), 68-74. This article is focussed on the role of László Rátz in the reform of mathematics education in Hungary, but it gives an excellent overview of this reform:**The reform of mathematics teaching.**

The various European reforms in mathematics teaching commenced in England. In early twentieth-century France, Poincaré and Langevin led the reform movement.

As for Germany, at the Congress of German Physicians and Nature-researchers in 1905, the Mathematics Reform Committee announced that the natural sciences have a cultural value apart from their practical applications, and, for this reason, their teaching should be placed on an equal footing with that of the philological disciplines. Worthy of special note is Felix Klein, professor at Göttingen University and an initiator and organiser of European reforms. Klein devoted special attention to the state of Hungarian mathematics. In 1905 he lectured in Budapest and over the course of the following decades converted Göttingen into something of a Mecca for Hungarian mathematicians.

Reforms in Hungary began with the efforts of Professor Manó Beke, a student of Klein's. At the annual meeting of the National Association of Secondary School Teachers in 1906, a Mathematics Reform Committee was created, with Beke as president, Sándor Mikola secretary, and László Rátz among the members. So thorough was the work of this committee that its results surpassed those of the committees elsewhere in Europe upon which it had been modelled, as was widely acknowledged both home and abroad. Testament to the quality of the Hungarian committee's activities was the appearance of its results in a book published by Teubner in 1911. Beginning in 1909, Beke, together with Rátz and Gusztáv Rados, represented Hungary on the International Reform Committee. László Rátz participated in congresses organised in Milan, Cambridge and Paris, and in 1910 was named "Officer d'Académie", a noteworthy French honour.

Even before the formation of the Hungarian committee, Rátz and Mikola had sensed the need for a change, and, inspired by the example of the English, they worked up a complete and highly detailed set of workable techniques and materials for the teaching of mathematics. They concluded that mathematics embraces a number of spontaneously acquired elements, which must be reinforced in the pupil. The study of mathematics must be interwoven with immediate experience. They stressed the importance of mental calculation and the practice of estimating. From their set of new materials and techniques they drew, by way of example, those sections devoted to the teaching of infinitesimal calculus and presented them in a number of publications: Rátz-Mikola, "Az infinitezimális számítás elemei a középiskolában" (Infinitesimal calculus in the secondary school), an article which appeared in the 1910 Annual Report of the Lutheran Gymnasium; in a book bearing the same title and appearing in the same year; and in a more sizeable volume entitled

*Functions and the elements of infinitesimal calculus*[A függvények és az infinitezimális számítások elemei], published by Franklin Publishers in 1914.

The most pressing questions connected with the reform of mathematics teaching were addressed in lectures at Congresses of the National Association of Secondary School Teachers. The material of these lectures soon found their way into a book edited by Beke and Mikola under the title

*The reform of mathematics teaching in the secondary school*[A középiskolai mathematikai tanítás reformja, Budapest, Franklin 1909]. Included in that book was an article by Rátz entitled "The teaching of functions and the elements of infinitesimal calculus in our secondary schools" [A függvények és az infinitesimális számítás elemeinek tanítása középiskoláinkban], in which he delivered the following summation:-

Beke, Mikola, Rátz and their collaborators took pains to point out that the reform was much more than a matter of expanding the content of the mathematics curriculum. They emphasised that the teaching of infinitesimal calculus must be centred upon the notion of functions. From the first form on, instruction should endeavour to shape the pupils' mode of reasoning and develop in them the capacity to think in terms of functions. "The content of mathematics instruction at the secondary level should be so prescribed as to include a place for the most essential notions of contemporary natural science". "We must transform the spirit of instruction, rather than simply tack on differential and integral calculus at the end of the curriculum".The principle behind the reform can be briefly expressed as follows: Let the teaching of mathematics be such as to develop in the pupil an awareness of the cultural importance of mathematics. We want the graduates of our secondary institutions to take something of their mathematical training into their adult lives. It is our hope that in such a way "the practice of thinking mathematically" will have an impact on public life. Our pupils must realise how great is the number of branches of mathematics which are related to practical life, the sciences and our general view of the world in its entirety. ... It is our conviction that teaching modified to this end is necessary in order to grasp the principal features of modern culture. It is not our goal to provide pupils bound for further technical and other specialised training with more mathematical knowledge; instead, we aim to equip precisely those pupils whose training in mathematics will come to an end upon their graduation from secondary school with an understanding of mathematics which is worthy of so great a science.

Pupils, they explained, must be able to reach a deeper understanding of reality through consideration of quantitative relations. In this respect, it was essential that the teacher strive to elucidate the instructional material with utmost clarity, as exemplified in their remarks on the teaching of the second and third forms: "... the content must be taught on the basis of a clear and vivid notion of the fraction".

In the course of the 1907-08 school year, the gymnasium, already equipped with a rich store of materials, sent to the cultural section of an exhibition in London a collection of tables and diagrams demonstrating the new method of mathematics teaching.

These multifaceted, courageous and path-breaking initiatives were crowned with success in November 1909 when the Lutheran Gymnasium was officially sanctioned to employ the new method of mathematics instruction in accordance with the stipulations of Rátz and Mikola, who, in fact, had been employing it on an experimental basis since 1902. In the course of the national educational reform of 1924, differential and integral calculus were introduced into the curriculum.