#4. How to Develop a Maths Culture
How high school maths competitions created a cluster of atomic bomb engineers
Over the years many competing theories have been proposed to try and explain why early 1900s Hungary— a small and obscure nation of just 7 million people—birthed some of the best mathematical and scientific geniuses history has ever seen.1 Whilst many of these articles have focussed on The Martians, they often overlook the man who initiated this upsurge in mathematical and scientific talent: Loránd Eötvös.23
Whilst Eötvös’ name is attributed to much of physics, from the Eötvös rule to the Eötvös number, and his name etched across Hungary’s institutions, his greatest contribution is arguably through his work at the Mathematical and Physical Society and the maths culture it spurted.4
So, who is he? What did he do? And why was the society he created so influential?
Loránd Eötvös
Born in 1848 to Agnes, a member of the illustrious Rosty de Barkóc family, and Jozsef, a writer and Minister of Education, Loránd, from receiving his doctorate until his death, was a physics professor at the University of Budapest.567 An important scientist in his own right, through his father he developed an interest in advancing the role science should play in wider society- strongly believing that superior knowledge of maths makes for better teachers and, therefore, a more advanced society.89
Because of this, in 1891 he founded an association of university professors and secondary school maths and physics teachers, known as the Mathematical and Physical Society. The goal:
"If we succeed in making every Hungarian teacher of mathematics and physics a real mathematician and physicist, then we not only serve the schools, but we raise the level of science in our country […] With dedication and seriousness […] in the future, new researchers and scientists [will] arise out of our group.”
To say he succeeded would be an understatement. The competition the society created would be won by almost all eminent Hungarian mathematicians and scientists we think of today, with many of these winners specifically crediting the competitions for developing a maths culture in Hungary and creating a self-fulfilling prophecy of excellence breeding excellence.10
So, how did it work?
In 1894, the society created a competition (today considered the first-ever maths Olympiad).11 Here, students of secondary school age compete against one another in a locked room for four hours and answer three maths questions that intend to test problem-solving ability and mathematical creativity, rather than factual recall, as is typical of exams.12 The 1913 winner, Tibor Rado, explained:
“The problems are selected […] in such a way that practically nothing, save one’s own brains, can be of any help […] the prize is not intended for the good boy; it is intended for the future creative mathematician.”
The judging panel looked not just for correctness, but imagination, elegance, clarity of language, and neatness- qualities that should serve mathematicians well in the future.
There were great incentives for students to compete: winners received a substantial sum of money, and their solutions were published in the Mathematical and Physical Journal. Crucially though, these winners were also granted automatic admission to the university of their choice.
Such a reward was particularly valuable during the times of numerus clausus, where restrictions on Jewish pupils, or those born in the ‘wrong class’ under communist rule, prevented them from attending university.13 For many Hungarian students, winning such a spot may have been their only path to higher education and working in their chosen field.14
Winning the competition resulted not only in personal pride for the student outperforming their peers, but was also a victory for the teacher and school- with the winners reminded that their achievements were only made possible partly by their teachers’ dedication and patience.15161718 In many cases, these teachers previously participated in, and indeed, even won the competition. Those teachers who were particularly dedicated (such as László Rátz and Lipót Fejér) coached several winners over the years, took great pride in doing so, and have received much acclaim from their students decades later.19202122
George Pólya recalls:
"Almost everybody of my age group was attracted to mathematics by Fejér.”
He would sit in a Budapest café with his students, writing formulas on menus, solving interesting maths problems and tell stories about mathematicians he had known. A whole culture developed around this man.23 Although his lectures were considered the experience of a lifetime, his influence outside the classroom was even more significant because it was around him that the first meaningful mathematical "school" developed in Hungary.
Years later, Pólya would develop his own unique teaching style—often referred to as the Pólya style. Perhaps this too was partially inspired by Fejér?
The prestige of the Eötvös Competition was well-known at the time and was something that Leo Szilard and John von Neumann discussed whilst working in Germany. Von Neumann even wrote to Fejér to ask for the names of the winners and runners-up because “Szilard is very interested in whether this procedure can be applied in the German context”. Von Neumann would then go on to ask Fejér’s opinion “about the extent to which the prizewinner and the talented are the same people.“24
It’s interesting that George Pólya, (who didn’t hand in his paper for the competition), and Gabor Szegő (the 1912 Eötvös winner), introduced a similar style of competition to California in 1946, with winners awarded a one-year scholarship to Stanford University. Unfortunately, despite receiving the posthumous backing of Theodore von Kármán, the programme didn’t last long and was discontinued in 1965.25 Pólya, recalls, "I told them it was a mistake to discontinue the program."
In the same year that the Eötvös Competition was founded, a Hungarian secondary school teacher named Daniel Arany, travelled to France, came across the Journal de mathématiques élémentaires, and became inspired to create his own maths journal.
The journal he founded (Középiskolai Matematikai Lapok, or KöMaL for short) also involved a competitive element. But, unlike the timed competitions used in the Eötvös Competition, KöMaL published its questions in a periodical, with new publications produced each month over an entire school year. In an average year, between 120 and 150 questions were published, with the journal receiving between 2500 and 3000 solutions from students.26
Here, greater emphasis was placed on careful thinking and stamina over speed.27 You were taught to value, not only the solution, but the best solution, the most beautiful solution: the model solution (minta vlilasz). Again, the goal was to develop a maths culture:
“We want to create a journal which will spread, in our dear homeland, day by day increasing scientific knowledge that will keep alive the interest of our mathematicians and physicists, to make science as well as the teaching of it a pleasurable experience.”
Just as the Eötvös Competition tried to emulate the work of real mathematicians by promoting creativity, neatness, and imagination, László Miklós Lovász, the son of Fields Medalist László Lovász, explains:
“In KöMaL, it matters how hard working you are. Some people are very good at competitions, but too lazy to do KöMaL. So it's very close to what a mathematician does, as far as I can tell.”
The incentives to compete were very similar to the Eötvös competition. Everybody who sent in correct solutions was listed by name, and the best solutions were printed in the journal. Frequent solvers had their pictures published at the end of the year. Over time, these people began to recognise one another. It was through KöMaL that Erdős would first meet his long-time collaborators Paul Turan and Tibor Gallai.
These are the headshots Turan and Gallai would have recognised when they first met Erdős:
Many educators worry that over-training for competitions hampers students’ independence and imagination, however, as George Pólya points out:
“If [the teacher] fills his allotted time with drilling his students in routine operations, he kills their interest, hampers their intellectual development, and misuses his opportunities. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.”
This was exactly the spirit of both the Eötvös and KöMaL competitions: they developed curiousity and independence, whilst also providing incentives to compete against—and ultimately beat— fellow students and teachers.28
Pólya recognises this in his book How to solve it?, as he references the importance of creativity and making discoveries at a young age:
“A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest: but if it challenges your curiosity and brings into play your inventive faculties, and you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.”
Testimonials
The legacy and impact of both competitions is well-documented amongst competitors, many of whom are now world-renowned mathematicians and scientists in their own right. Here are a few additional testimonials from eminent mathematicians detailing the importance of KöMaL and the Eötvös Competition in creating a maths culture.
In response to the question: Do you feel that your mathematical development was affected by KöMaL? Paul Erdős, one of the most prolific mathematicians and producers of conjectures of the 20th century, replied:
“Yes, of course. You actually learn to solve problems there. And many of the good mathematicians realise very early that they have ability.”
When George Pólya was asked to explain the Hungarian phenomenon, he attributed it to three factors: KöMaL, the Eötvös Competition, and Fejér:
"The journal [KöMaL] stimulated interest in mathematics and prepared students for the Eötvös Competition."
“The [Eötvös] competition created interest and attracted young people to the study of mathematics."
"[Fejér] was responsible for attracting many young people to mathematics, not only through formal lectures but also through informal discussions with students."
Agnes Berger, described KöMaL as: “tremendous entertainment” and Tibor Gallai recalls:
“Nowhere else in the world is there this kind of high school paper [KöMaL], and this more than anything else is responsible for the excellence of Hungarian mathematics.”
Gábor Szegő, the 1912 Eötvös winner, and John von Neumann’s tutor
“remember[s] vividly the time when I participated in this […] journal [...] I would wait eagerly for the arrival of the monthly issue and my first concern was to look at the problem section, almost breathlessly, and to start grappling with the problems without delay. The names of the others who were in the same business were quickly known to me, and frequently I read with considerable envy how they had succeeded with some problems which I could not handle with complete success, or how they had found a better solution (that is, simpler, more elegant or wittier) than the one I had sent in.”
Peter Hermann, an editor of KöMaL:
“This can be stated for sure: Almost everyone who became a famous or nearly famous mathematician in Hungary, when he or she was a student, they took part in this contest [KöMaL]. I actually personally do not know anyone among them who would be a counterexample to that statement”.
Sources
Connelly, J., (2010), A tradition of excellence transitions to the 21 st century: Hungarian mathematics education, 1988-2008, Columbia University.
Connolly Stockton, J., (2012), Mathematical Competitions in Hungary: Promoting a Tradition of Excellence & Creativity, TME, vol. 9, No 1&2, p37.
Csapó, B., (1991), Math Achievement in Cultural Context: The Case of Hungary.
Gosztonyi, K., (2016), Mathematical Culture and Mathematics Education in Hungary in the XXth Century, Mathematical Cultures, pp 71–89.
Hersh, R., and John-Steiner, V., (1993), A visit to Hungarian mathematics, The Mathematical Intelligencer, Vol.15, pp 13–26.
Horvath, J., (2006), A Panorama of Hungarian Mathematics in the Twentieth Century, Bolyai Society Mathematical Studies, Volume 14.
Kenderov, P. S., (2006), Competitions and mathematics education, Proceedings of the International Congress of Mathematicians, Vol. 3.
Kenderov, P. S., (2009), A short history of the World Federation of National Mathematics Competitions, Mathematics Competitions, Vol. 22, Issue 2, pp14-31.
Tunde Kantor-Varga, A Panorama of Hungarian Mathematics in the Twentieth Century, Bolyai Society Mathematical Studies, Vol. 14, pp. 563–607.
Wieschenberg, A. A., and Polya, G., A (1987), Conversation with George Polya, Mathematics Magazine, vol. 60, no. 5, pp. 265–68.
Wieschenberg, A. A., (1990), The Birth of the Eötvös Competition, The College of Mathematics Journal, Vol. 21, No 4, p286-293.
Lipót Fejér by those who knew him, St Andrews University
Lóránd Baron von Eötvös, St Andrews University
Mathematical and Physical Journal for Secondary Schools, St Andrews University
The Martians and Their Schooling by ‘Thomas Jefferson Snodgrass’ (pseudonym)
Interesting articles and videos:
See here for a list of the questions asked in the Eötvös Competition throughout history.
I highly recommend watching this video of Edward Teller describing the genius of John von Neumann. The clip is taken from this hour-long film
The Martians and their Schooling - a very good summary of the Hungarian Phenomenon.
This article by Jørgen Veisdal in the excellent Privatdozent Substack details more about the life of Paul Erdős. This one focusses on The Martians, and this one on John von Neumann.
Excluding the Bolyai father and son, it is notable that Hungary did not produce any notable outstanding mathematical talent before the turn of the century. However, this would change:
“In the period of the two decades around the First World War, [Budapest] proved to be an exceptionally fertile breeding ground for scientific talent.” [Source: Stanisław Ulam]
Today, Hungary is cited as producing the largest per capita number of mathematicians and physicists in the first half of the 20th century. [Source: Connelly Stockton, 2010]
‘The Martians’ are the collective of Jewish Hungarian mathematicians who emigrated to America in the wake of the Second World War (Theodore von Karman, Leo Szilard, Eugene Wigner, Edward Teller, and John von Neumann).
In his book, The Voice of the Martians, György Marx recounts the tale of how they got their name:
“The universe is vast, containing myriads of stars[...] likely to have planets circling around them [...] The simplest living things will multiply, evolve by natural selection and become more complicated till eventually active, thinking creatures will emerge […] Yearning for fresh worlds […] they should spread out all over the Galaxy. These highly exceptional and talented people could hardly overlook such a beautiful place as our Earth. "And so," Fermi came to his overwhelming question, "if all this has been happening, they should have arrived here by now, so where are they?" – It was Leo Szilard, a man with an impish sense of humour, who supplied the perfect reply to the Fermi Paradox: "They are among us," he said, "but they call themselves Hungarians."
Tibor Gallai, in response to the questions: “Do you have any idea why this took place in Hungary? What was it in this country that made this possible?” replied:
“For part of 1894 and 1895 the Minister of Education was Lorand Eötvös, after whom the University is named. He was deeply committed to the development of Hungarian culture and science. While he was in office there was founded the Eötvös Collegium, with the purpose of improving the training of high school teachers. So he is part of what stimulated our development.”
Institutions named after Eötvös, include the Eötvös Loránd Physics Society; the Eötvös Loránd Geophysical Institute; and the Eötvös Loránd University (ELTE). Eotvos is also used as a unit of gravitational gradient, and his name is used to describe both the Eötvös effect and the Eötvös experiment.
At the time of Lóránd's birth, his father was minister of education in the revolutionary government of 1848. However, after a disagreement with Governor-President Lajos Kossuth, he resigned later in the year and went to Munich where he lived until 1851.
Eötvös' father died on 2 February 1871. In the same year, Loránd returned to Hungary and was appointed to the University of Budapest as a privatdozent and promoted to full professor of theoretical physics. He had now inherited his father's ‘Baron’ title and became a member of the Upper House in the Hungarian Parliament in 1872.
From June 1894 to January 1895 Eötvös was the minister of public instruction in the Cabinet of Sandor Wekerle. Here, he founded the Eötvös Collegium to improve the teaching of Hungarian secondary school teachers. He resigned from this government post just one year later to devote himself fully to teaching physics at the University of Budapest [Source].
At the start of the Industrial Revolution, József Eötvös (then the minister of culture), decided to introduce secular schools. He asked Mór Kármán to do the job, and he created the Minta Gymnasium in Budapest, made famous by its future students Theodore von Kármán, Edward Teller, Michael Polanyi and Peter Lax.
Lorand Eötvös taught George Pólya, and, according to Pólya was the best teacher at the University [Source].
This is shown in letters that they wrote to one another. On 28 March 1866, before his trip abroad, Loránd Eötvös wrote to his father:
“Ambition and a sense of duty […] are my inborn qualities, [and] to satisfy these two impulses while maintaining my individual independence: this is my life goal, and at least so far, I have found that I can best fulfil it by entering a scientific career.”
A year later, on 24 March 1867, when his career choice seemed, for the time being decided, Lorand’s father wrote back:
“It is a real comfort to me to see you in another career. Go forward with courage and don’t regret your efforts. The greatest efforts in science will be rewarded, because it is not expected from men, but is found in science itself.” [Source]
His work on the weak equivalence principle played a prominent role in the theory of relativity and was cited by Einstein in his 1913 paper [Source].
Notable winners include those who worked at the Manhattan Project to develop the atomic bomb, Theodore von Karman (1898) and Edward Teller (1925), as well as John von Neumann’s tutors: Lipot Fejér (1897) and Gábor Szegő (1912).
Kenderov (2009) briefly mentions an earlier maths competition taking place in Bucharest on the 21st of May 1885, in which 70 students competed, with 2 girls and 9 boys receiving prizes. However, very little information beyond this is known.
Students were allowed to bring in books and notes.
In the first 11 years, between 49 and 98 students took part each year, with approximately 29 to 56 actually handing in their papers. [Source: Wieschenberg, and Polya, 1987]
“The flowering of Hungarian talent in the generation of the cultural wave was due to the special social and cultural circumstances existing in Hungary at the turn of the century. By then a strong middle class had emerged and asserted itself. Having risen in response to needs that the nobility did not feel inclined to fill and the peasants could not fill, it was largely Jewish and was animated by the intellectual ambitions of the Jews. The intellectual portion of this middle class converged upon the capital where it created a peculiarly sophisticated atmosphere, and kept its members under continuous stimulation. The political anti-Semitism of the early twenties hit this segment of the population with great vehemence and gave the intellectuals a further reason for striving to excel and stay afloat. Under these circumstances, talent could not remain latent. It flourished.” - [Source: Laura Fermi, cited in Hersh and John-Steiner, 1993]
“Fin de siècle Budapest was a place of wealth, modernity, and learning. It was home to 1 million souls on the eve of World War I, 20% of whom were Jewish. The Christian gentry in the city had jobs in government and law, but commerce, finance, and engineering were seen as beneath them, and therefore wide open to the Budapest Jews.
For the Jews there was a heavy cultural emphasis on scholarship. Fasori Evangélikus Gimnázium, the school that both Wigner and von Neumann attended, charged Jews five times the tuition fee that Lutherans paid, and yet the Jews came anyways.
The power of these schools wasn’t merely a consequence of pedagogy or teachers, it was a product of the culture in which they were rooted. Gymnasium students formed "self-improvement circles” of their own volition, giving talks to each other on topics outside the curriculum.” [Source]
Eötvös’ motto was "Learn from each other so that you can teach better" [Source: Connelly, 2010]
The emphasis on creativity, competition between students, and collaboration between teachers and students was also replicated in the school system. Here is Theodore von Karman talking about his school:
“Mathematics was taught in terms of everyday statistics: We looked up the production of wheat in Hungary, set up tables, drew graphs, learned about the ‘rate of change’ which brought us to the edge of calculus. At no time did we memorise rules from a book. Instead, we sought to develop them ourselves. The Minta was the first school in Hungary to put an end to the stiff relationship between the teacher and the pupil which existed at that time. Students could talk to the teachers outside of class and could discuss matters not strictly concerning school. For the first time in Hungary, a teacher might go so far as to shake hands with a pupil in the event of their meeting outside of class.” [Source: Hersh and John-Steiner, 1993]
“The Hungarian gimnázium education at the beginning of the twentieth century was an elitist kind of education, rather broad based in its curriculum, with a select group of students, and highly educated and cultured teachers who were members of an appreciated and respected profession. The excellence of the Budapest high schools came partly from the fact that they catered to a small portion of the population—high school education at that time was not yet for everyone. To be a high school teacher meant prestige in society and life at a comfortable level.” [Source]
“Our gimnázium teachers had a vital presence,” recalled Wigner. “To kindle interest and spread knowledge among the young—this was what they truly loved.”
In 1905, Henri Poincaré came to Budapest to accept the first Bolyai prize. When he got off the train he was greeted by high-ranking ministers and secretaries (possibly because he was a cousin of Raymond Poincare, (the politician who later became President and four times Premier of the Third Republic). He looked around and asked, "Where is Fejér?" The ministers and secretaries looked at each other and said, "Who is Fejér?". Poincaré responded, "Fejér is the greatest Hungarian mathematician, one of the world's greatest mathematicians."
Within a year, Fejér was a professor in Kolozsvar. Five years later, mainly by Lorand Eötvös’ intervention, he was offered a chair at the University of Budapest. [Source: Hersh and John-Steiner, 1993]
Many young talented people were attracted into Fejér’s circle through his personality, including Alfréd Haar, Theodore von Kármán, Dénes König, Gábor Szegő, George Pólya and Paul Erdős.
“Fejér gave very short, very beautiful lectures. They lasted less than an hour. You sat there for a long time before he came. When he came in, he would be in a sort of frenzy. He was very ugly-looking when you first examined him, but he had a very lively face with a lot of expression. The lecture was thought out in very great detail, with dramatic denouement. He seemed to relive the birth of the theorem; we were present at the creation. He made his famous contemporaries equally vivid; they rose from the pages of the textbooks. That made mathematics appear as a social as well as an intellectual activity.” (Source: Ágnes Berger, cited in Hersh and John-Steiner, 1993]
“There was hardly an intelligent, let alone a gifted student who could exempt himself from the magic of his lectures’, says fellow Hungarian mathematician George Polya. ‘They could not resist imitating his stress patterns and gestures, such was his personal impact upon them. Fejér’s interest in his young charges when well beyond the call of duty. ‘What does little Johnny Neumann do?’ he would write to Szegő years later.” [Source, pp11-12]
Eugene Wigner recalls:
“My own history begins in the high school in Hungry where my mathematics teacher, [László] Rátz, gave me books to read and evoked in me a sense of the beauty of his subject.”
“He had every quality of a miraculous teacher: He loved teaching. He knew the subject and how to kindle interest in it. He imparted the very deepest understanding. [...] No one could evoke the beauty of the subject like Rátz
Apparently, in the late 1970s, 60 years after leaving his high school, Wigner was asked whether he remembers Rátz, and Wigner replied, “There he is!” and pointed to a photograph of Rátz on his office wall.
From 1894 to 1914, Ratz served as editor-in-chief of KöMaL.
“He edited the journal for twenty years, and did so with the utmost selflessness, receiving no support from the state or any other source (not that he asked for it) and, indeed, contributing substantial amounts of his own money to ensure the journal's publication” [Source]
“The hours spent in continental coffee houses with Fejér discussing mathematics and telling stories are a cherished recollection for many of us” - George Pólya
The letter in full:
Dear highly honored Professor,
I had the opportunity several times to speak to Leo Szilard about the student competitions of the Eötvös Mathematical and Physical Society, also about the fact that the winners of these competitions, so to say, overlap with the set of mathematicians and physicists who later became well-respected world-figures. Taking the general bad reputations of examinations world-wide into account it is to be considered as a great achievement if the selection works with a 50 percent probability of hitting the talent. Szilard is very interested in whether this procedure can be applied in the German context and this has been the subject of much discussion between us. However, since we would like above all to learn what the reliable statistical details are, we are approaching you with the following request. We would like:
1. To have a list of names of the winners and runners-ups of the student competitions,
2. To see marked on the list those who were adopted on a scientific basis and those adopted for other work,
3. To know your opinion about the extent to which the prizewinner and the talented are the same people and, for example, what proportion of the former would be worthy of financial support from the State in order to make their studies possible.
“I tried out for this prize against students of great attainments, and to my delight I managed to win. Now, I note that more than half of all the famous expatriate Hungarian scientists, and almost all the well-known ones in the United States, have won this prize. I think that this kind of contest is vital to our educational system, and I would like to see more such contests encouraged…in the United States and in other countries.” - Source: Theodore von Karman cited in Hersch and John-Steiner, 1993]
Source: Kenderov (2006)
Proponents argue that this is good practice for how mathematicians actually work.
Although there may be growing discomfort with the competition-driven mindset of the Hungarian mathematics education system and concern that it is becoming too much of a “sport” rather than studying mathematics, the continued emergence of new contests suggests that competition itself remains a key way to engage students and motivate their interest in mathematics [Source].