by Mark Saul

This summer marks the thirtieth year since the end of the Soviet Union. It also marks the passing of one of the great figures of Russian mathematical culture, Nicholas Nikolayevich Konstantinov. This note concerns both events, but cannot do justice to either. Rather, I will here give some personal reminiscences that might contribute to the picture, but not find a place in the historical record. I leave to other sources the task of a more comprehensive account. Here’s my story.

The year was 1987. The Cold War was still smoldering, but no longer raging. I received a phone call from an American teaching colleague: “I got an email message for you from one Professor Konstantinov in Moscow.”

Just the fact that this message had arrived was remarkable. The World Wide Web had yet to appear. Email was new and laptops rare. And it was not yet clear that the internet could form a bridge between the two camps in the global political stalemate. Was a graph of the Eastern and Western computer networks even connected? How did a colleague from Moscow contact me? And why?

I had known for a long time about the remarkable flowering of mathematics in the USSR in the postwar years. Reading Russian, I had access to materials in that language: a subscription to *Kvant (* http://www.kvant.info/ ; http://kvant.mccme.ru/), the journal for pre-college students started by Kolmogorov and other scientists. I also subscribed to *Matematika v Shkole* (http://www.schoolpress.ru/products/magazines/index.php?SECTION_ID=42&MAGAZINE_ID=34945), a journal for teacher of mathematics.

Finally, I possessed a hard-earned personal collection of Russian texts and problem books. Hard-earned, because at that time one had to subscribe in advance to Soviet books that would then ‘be published in future, through a newsletter that listed all such books. I would comb through it weekly and order the ones that looked interesting. Sometimes they came, and sometimes they were of great interest. And sometimes I got a book in Hindi, or a treatise on diseases of cattle. Luckily, the books were uniformly inexpensive. I had access only through the newsletter, *Noviye Knigi SSSR *[New books from the USSR], not through recommendations from working mathematicians or teachers. Through these sources, I learned of the work of Konstantinov and his colleagues.

So what did Nikolai Konstantinov want? That was not clear. Mostly, he wanted to make contact. He had heard of my work, and I of his, through various meetings and publications. I replied to his email, but it was a while before the next contact.

In 1989, I was at an international meeting in Waterloo, Ontario, when someone slid into the seat next to mine and addressed me, in Russian. “What is he saying?” the man inquired. I whispered back a short summary, wondering why he had somehow assumed I would understand him. He replied, as if reading the confusion in my mind, “I am Konstantinov.” He had already read my own name badge.

We sat through the presentations—none of them in Russian or German (Konstantinov’s second language). Then we sat around and over dinner talked about our work, about the difficulties in each of our environments, and about possibilities of collaboration.

Our next contact was in 1990, which was to be the last full year of the existence of the USSR. The Iron Curtain had lifted from Eastern Europe, and many thought that Russia would be next. I got a call directly this time, from someone in Brooklyn. By then a large community of Russian emigrés, mostly Jewish, had settled in New York, and were about to make a significant impact on the math education scene. But not quite yet.

The person who called me was one Irina Speranskaya, who worked in Moscow for a government agency in the nascent area of trade with America. She was visiting New York, and brought me a new book from Konstantinov, with an offer. If I came to Moscow with a group of American students, we would have a Russian-style summer camp, with all expenses (once we arrived in Moscow) borne by them. Miraculously, the National Science Foundation was willing to fund the trip, and I found myself leading 20 US students and five teachers to Moscow for an immersion in Russian pre-college mathematics. I have written accounts elsewhere of this trip (Saul 1992), which contributed to the introduction of a number of Russian cultural traditions to the US.

Well, the Soviet Union fell—just two weeks after the conclusion of the NSF-sponsored summer program—and communication with Russian teachers and mathematicians became more and more common. My personal experience was duplicated by many others. More American teachers and mathematicians came in contact with Russian emigrés and started programs—math circles and math camps—inspired by their experience. And today, if you are reading this essay in the math department of any American university, you can probably walk down the hall and check its accuracy with a colleague who grew up in Russia or the USSR.

Part of what I discovered in the waning USSR, and which Russian mathematicians had long understood, was the remarkable nature of the Soviet mathematical community. It was more of a subculture than a community, or even a counter-culture to the official totalitarian ideology. The origins and characteristics of this phenomenon have been described in detail elsewhere (Gerovitch 2013, Karp 2010, Polyakova 2010, Sossinsky 2010). In brief, the Soviet government attempted to control intellectual life in the country. The arts were heavily, and famously, politicized—even music, perhaps the most abstract of artistic communications. This control was direct and could be brutal. Control of the sciences was often subtler. Certain lines of investigation were encouraged, others stifled. Advances in genetics and computer science, perhaps the two most exciting branches of science in the post-war era, were discouraged or even prohibited. The social sciences were likewise politicized. The physical sciences were largely put at the service of the military (Chan 2015). But even in more academic environments, the need for laboratory facilities was a powerful lever of control.

Mathematics, on the other hand, offered a refuge. One needed no equipment and was given little direction. The applications of one’s work were often sufficiently far removed from the work itself to make the connection between the two difficult for outsiders to fathom. So active minds flocked to mathematics, minds which could have found occupation in other areas had totalitarian forces not been at work. Doing mathematics even became an act of rebellion, of silent refusal to honor the needs of the government. And all this could happen without physical or verbal expression, just by acting as mathematicians or students of mathematics.

So, for example, the social and professional lives of a mathematician were often the same. Summer camps for students, after-school math circles and study groups, all became part of a tradition of enjoying mathematics as people were pushed together by the sometimes harsh totalitarian intellectual climate.

Konstantinov was both a product and a bearer of this unique mathematical culture. I offer here just a few glimpses, from personal recollection, of how it felt to be a part of it.

After the 1991 summer camp, I was invited to to the summer seminar of the International Tournament of the Towns (https://www.turgor.ru/en/). This involved traveling 30 hours by train across the vastness of Russia to Chelyabinsk, the first big city on the Siberian side of the southern Ural Mountains.

This trip was memorable in numerous ways. Konstantinov regaled us with tales of people and events he had known, or known of. There was the mathematician who was the son of a pre-revolutionary railroad magnate, and who recalled traveling around Russia in his youth on a private company railroad car. There was the tale of the runaway train, on the very tracks we were traversing, which rolled from the top of the Ural pass miles down to more inhabited areas. A locomotive was sent to chase and capture it. The locomotive collided forcefully with the train from the rear, to couple with and stop it. The collision was enormous, but prevented the train from devastating a more populated area. We were traveling through the Bashkir Republic, and these Turkic people had a heritage of horsemanship. Konstantinov challenged us to spot a rider on horseback. But all we could see was pipelines from now-exhausted oil wells. Each tale was more interesting than the last, and contributed to a picture of the country and of its mathematical community that few people have glimpsed who have not grown up there.

And the mathematics! We talked for two days about math problems. About ways to classify them. About which were suited for competition and which were not. About logical riddles and their relationship to mathematics. About how contest problems sometimes ended up applied. Three samples stand out in my memory:

An Olympiad problem had been set by Alexey Kanel-Belov a few years before, about packing polyhedra so that their cross-sections tessellated a plane. It turns out that for some such tessellations, the polyhedral blocks forming it will hold each other up when the configuration is lifted. A student solved this problem, and brought it home to his father, an engineer. The father then used it to design tilings for ceilings. (See also Kanel-Belov 2008.)

We discussed a problem about a wire frame forming a cube. Consider the edges as segments. If it is to pass through a plane, what is the smallest length slit you must cut in the plane? That is, suppose the wire frame grew very hot, and had to pass through a piece of paper. What is the smallest ‘length’ of paper that must be burned? This was an interesting problem, but how would the contestants express their solution? They would have to describe the motion of the cube as it passes through the plane. Some motions, even in two dimensions, are difficult to describe. But in three dimensions? We decided not to use this problem.

A third problem was about the “Devil’s Staircase,” a now-classic way of using the Cantor set to define a step function which is continuous. It was decided that there is enough here to offer students who have not had an introduction to analysis. The analytic implications of the results could be appreciated as they learned more.

The reason for this seminar-on-wheels lay in the traditions of Soviet mathematics. In the USSR, teachers had very limited access to copying machines of any sort. Among other reasons, these could be used to reproduce unauthorized literature and so worked against control of information by the state. So test questions had to be written on the blackboard or even given orally. This led to traditions in testing—and in contests—which emphasized long-answer ‘Olympiad’ style problems, rather than the short answer problems more typical of American competitions. Many competitions included rounds that were conducted completely orally (for example, see Fomin and Kirichenko, 1994) . And the tradition of math battle or math wrangle (https://www.maa.org/sites/default/files/pdf/sections/math_wrangle.pdf) also evolved partly from this circumstance.

And in fact we were responsible for setting the problems of a math wrangle at the seminar in Chelyabinsk, a gathering of local winners of the Tournament of the Towns. Later, at the camp itself, I witnessed the process of giving the students the problems, a process very different from any American contest I have known. The contestants gathered in a room, and the judges wrote the problem statements on a chalk board. They then explained the problems orally, taking questions from the audience to make sure the problem statements were clear. Finally the students were given three days to solve the problems and present them in math wrangle format.

The Tournament of the Towns was a child of the fertile brain of Konstantinov, offered as an alternative to the rapidly rigidifying structures leading to the International Mathematical Olympiad. Competition is by ‘town’ (city). In the tradition of Russian/Soviet competitions, problems all require solutions written out, and are selected so as to include both novice problem solvers and those with sophisticated background.

Konstantinov’s work was central to numerous other initiatives. In 1978, he started the Lomonosov Tournament, a multi-subject competition named after Mikhail Lomonosov, the 18^{th} century polymath considered by many to be the father of Russian academia. This tournament has been held every year since. In 1990 Konstantinov was one of the founders of the Independent University of Moscow, among the leading institutions of higher learning in mathematics in Russia. And well into his later years, Konstantinov continued working in Moscow High School 179, and helped to edit *Kvant* magazine. Matusov (2017) gives an account of his fresh approach to the classroom, as well as another set of personal reminiscences of Russian/Soviet mathematical culture.

On one of my visits to Moscow, I was fortunate enough to catch a talk by the Russian mathematician Evgeniy Dynkin, visiting Moscow from his position Cornell University. The talk was for high school students, and the topic was a classic problem in probability: A sequence of integers is presented to you, one at a time, then each disappears. You must choose the largest you can. After your choice, the integers stop coming. (This is a model for numerous life experiences—even for high school students–from choosing a spouse or date to finding lodging along a highway.) In classic Russian style, Dynkin was able to break the problem down for his audience. I had seen this sort of exposition before, and was not surprised. What struck me, however, was the collegiality between Dynkin and Konstantinov. They spoke together, both before and after the presentation, about the level of the students, about how the presentation had gone, and about various mathematical and educational activities going on in Moscow. They were clearly members of the same community. It is now becoming more common to find such camaraderie in the American mathematical community.

After the fall of the USSR, when Russians had the opportunity to travel abroad, Konstantinov and I worked together in various places around the world. I recall him balancing on a beam which lay precariously across a swimming pool in Canada. He and I shopped for souvenirs in Australia, where he bought tiny koalas for each student in one of his classes. And in Amman, Jordan, we sat down to dinner at a conference we were both attending. The dinner plates were square. Konstantinov challenged me to find a reason for this shape. His reason? To make it easier to calculate their area.

Konstantinov’s humor, his fresh attitude towards learning, his creative structuring of programs… they will all be missed, even as his legacy continues.

REFERENCES

Chan, Chi Ling (2015). Fallen Behind: Science, Technology, and Soviet Statism. Intersect, 8(3) (1-11). http://ojs.stanford.edu/ojs/index.php/intersect/article/view/691.

Fomin, D., and Kirichenko, A. (1994) *Leningrad Mathematical Olympiads 1987-1991*. Westford, MA: MathPro Press.

Gerovitch, S, (2013). Parallel Worlds: Formal Structure and Informal Mechanisms of Postwar Soviet Mathematics, *Historia Scientiarum*, 22(3),181-200. https://www.academia.edu/5366902

Kanel-Belov, Alexey et al. (2010) Interlocking of Convex Polyhedra: towards a Geometric Theory of Fragmented Solids. Moscow Mathematical Journal., 10:2, 337–342, 2010 (https://arxiv.org/abs/0812.5089 ).

Karp, A. (2010). Reforms and Counter-Reforms: Schools between 1917 and the 1950s, in Karp, A., and Vogeli, B. (eds.) *Russian Mathematics Education: History and World Significance*, Singapore: World Scientific Publishing Co. (43-86)

The Lomonosov Tournament, 1996, *Math. Ed.*, 1997, Issue 1, 79–106 (in Russian) http://www.mathnet.ru/links/075412a9f93379a3b310240dede3b677/mo232.pdf

Matusov, Eugene (2017) Nikolai N. Konstantinov’s Authorial Math Pedagogy for People with Wings, *Journal of Russian & East European Psychology,* 54:1, 1-117, DOI: 10.1080/10610405.2017.1352391 http://dx.doi.org/10.1080/10610405.2017.1352391

Polyakova, T. (2010) “Mathematics Education in Russia before the 1917 Revolution”, in Karp, A., and Vogeli, B. (eds.) *Russian Mathematics Education: History and World Significance*, Singapore: World Scientific Publishing Co. (1-42)

Saul, M. (1992). Love Among the Ruins. *Focus, 12*(1), 1,6,7, https://www.maa.org/sites/default/files/pdf/pubs/focus/past_issues/FOCUS_12_1.pdf. Accessed June 2020.

Sossinsky, A. (2010) “Mathematicians and Mathematics Education: A Tradition of Involvement”, in Karp, A., and Vogeli, B. (eds.) *Russian Mathematics Education: History and World Significance*, Singapore: World Scientific Publishing Co. (187-222)