by Mark Saul<\/p>\n
This summer marks the thirtieth year since the end of the Soviet Union.\u00a0 It also marks the passing of one of the great figures of Russian mathematical culture, Nicholas Nikolayevich Konstantinov.\u00a0 This note concerns both events, but cannot do justice to either. \u00a0Rather, I will here give some personal reminiscences that might contribute to the picture, but not find a place in the historical record.\u00a0 \u00a0\u00a0I leave to other sources the task of a more comprehensive account.\u00a0 Here\u2019s my story.<\/p>\n
The year was 1987.\u00a0 The Cold War was still smoldering, but no longer raging.\u00a0 I received a phone call from an American teaching colleague: \u201cI got an email message for you from one Professor Konstantinov in Moscow.\u201d<\/p>\n
Just the fact that this message had arrived was remarkable.\u00a0 The World Wide Web had yet to appear.\u00a0 Email was new and laptops rare.\u00a0 And it was not yet clear that the internet could form a bridge between the two camps in the global political stalemate.\u00a0 Was a graph of the Eastern and Western computer networks even connected?\u00a0 How did a colleague from Moscow contact me?\u00a0 And why?<\/p>\n
<\/p>\n
I had known for a long time about the remarkable flowering of mathematics in the USSR in the postwar years.\u00a0 Reading Russian, I had access to materials in that language: a subscription to Kvant (<\/em> http:\/\/www.kvant.info\/<\/a> ; http:\/\/kvant.mccme.ru\/<\/a>), the journal for pre-college students started by Kolmogorov and other scientists.\u00a0 I also subscribed to Matematika v Shkole<\/em> (http:\/\/www.schoolpress.ru\/products\/magazines\/index.php?SECTION_ID=42&MAGAZINE_ID=34945<\/a>),\u00a0 a journal for teacher of mathematics.<\/p>\n Finally, I possessed a hard-earned personal collection of Russian texts and problem books.\u00a0 Hard-earned, because at that time one had to subscribe in advance to Soviet books that would then \u2018be published in future, through a newsletter that listed all such books.\u00a0 I would comb through it weekly and order the ones that looked interesting.\u00a0 Sometimes they came, and sometimes they were of great interest.\u00a0 And sometimes I got a book in Hindi, or a treatise on diseases of cattle.\u00a0 Luckily, the books were uniformly inexpensive.\u00a0 I had access only through the newsletter, Noviye Knigi SSSR <\/em>[New books from the USSR], not through recommendations from working mathematicians or teachers.\u00a0 Through these sources, I learned of the work of Konstantinov and his colleagues.<\/p>\n So what did Nikolai Konstantinov want?\u00a0 That was not clear. Mostly, he wanted to make contact.\u00a0 He had heard of my work, and I of his, through various meetings and publications.\u00a0 I replied to his email, but it was a while before the next contact.<\/p>\n 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?”\u00a0 the man inquired.\u00a0 I whispered back a short summary, wondering why he had somehow assumed I would understand him. \u00a0He replied, as if reading the confusion in my mind, \u201cI am Konstantinov.\u201d\u00a0\u00a0 He had already read my own name badge.<\/p>\n We sat through the presentations\u2014none of them in Russian or German \u00a0(Konstantinov\u2019s second language).\u00a0 Then we sat around and over dinner talked about our work, about the difficulties in each of our environments, and about possibilities of collaboration.<\/p>\n Our next contact was in 1990, which was to be the last full year of the existence of the USSR.\u00a0 The Iron Curtain had lifted from Eastern Europe, and many thought that Russia would be next.\u00a0 I got a call directly this time, from someone in Brooklyn.\u00a0 By then a large community of Russian emigr\u00e9s, mostly Jewish, had settled in New York, and were about to make a significant impact on the math education scene.\u00a0 But not quite yet.<\/p>\n 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.\u00a0 She was visiting New York, and brought me a new book from Konstantinov, with an offer.\u00a0 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.\u00a0 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.\u00a0 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.<\/p>\n Well, the Soviet Union fell\u2014just two weeks after the conclusion of the NSF-sponsored summer program\u2014and communication with Russian teachers and mathematicians became more and more common.\u00a0 My personal experience was duplicated by many others.\u00a0 More American teachers and mathematicians came in contact with Russian emigr\u00e9s and started programs\u2014math circles and math camps\u2014inspired by their experience.\u00a0 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.<\/p>\n 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.\u00a0 It was more of a subculture than a community, or even a counter-culture to the official totalitarian ideology.\u00a0 The origins and characteristics of this phenomenon have been described in detail elsewhere \u00a0(Gerovitch 2013, Karp 2010, Polyakova 2010, Sossinsky 2010).\u00a0 In brief, the Soviet government attempted to control intellectual life in the country.\u00a0 The arts were heavily, and famously, politicized\u2014even music, perhaps the most abstract of artistic communications.\u00a0 This control was direct and could be brutal.\u00a0 Control of the sciences was often subtler. Certain lines of investigation were encouraged, others stifled.\u00a0 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).\u00a0 But even in more academic environments, the need for laboratory facilities was a powerful lever of control.<\/p>\n Mathematics, on the other hand, offered a refuge.\u00a0 One needed no equipment and was given little direction.\u00a0 The applications of one\u2019s work were often sufficiently far removed from the work itself to make the connection between the two difficult for outsiders to fathom.\u00a0 So active minds flocked to mathematics, minds which could have found occupation in other areas had totalitarian forces not been at work.\u00a0 Doing mathematics even became an act of rebellion, of silent refusal to honor the needs of the government.\u00a0 And all this could happen without physical or verbal expression, just by acting as mathematicians or students of mathematics.<\/p>\n So, for example, the social and professional lives of a mathematician were often the same.\u00a0 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 \u00a0harsh totalitarian intellectual climate.<\/p>\n Konstantinov was both a product and a bearer of this unique mathematical culture.\u00a0 I offer here just a few glimpses, from personal recollection, of how it felt to be a part of it.<\/p>\n 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\/).\u00a0 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.<\/p>\n This trip was memorable in numerous ways.\u00a0 Konstantinov regaled us with tales of people and events he had known, or known of.\u00a0 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.\u00a0 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.\u00a0 A locomotive was sent to chase and capture it.\u00a0 The locomotive collided forcefully with the train from the rear, to couple with and stop it.\u00a0 The collision was enormous, but prevented the train from devastating a more populated area.\u00a0 We were traveling through the Bashkir Republic, and these Turkic people had a heritage of horsemanship.\u00a0 Konstantinov challenged us to spot a rider on horseback.\u00a0 But all we could see was pipelines from now-exhausted oil wells.\u00a0 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.<\/p>\n And the mathematics!\u00a0 We talked for two days about math problems.\u00a0 About ways to classify them.\u00a0 About which were suited for competition and which were not. About logical riddles and their relationship to mathematics.\u00a0 About how contest problems sometimes ended up applied.\u00a0 Three samples stand out in my memory:<\/p>\n An Olympiad problem had been set by Alexey Kanel-Belov \u00a0a few years before, about packing polyhedra so that their cross-sections tessellated a plane.\u00a0 It turns out that for some such tessellations, the polyhedral blocks forming it will hold each other up when the configuration is lifted. \u00a0A student solved this problem, and brought it home to his father, an engineer.\u00a0 The father then used it to design tilings for ceilings. (See also Kanel-Belov 2008.)<\/p>\n We discussed a problem about a wire frame forming a cube.\u00a0 Consider the edges as segments.\u00a0 If it is to pass through a plane, what is the smallest length slit you must cut in the plane?\u00a0 That is, suppose the wire frame grew very hot, and had to pass through a piece of paper.\u00a0 What is the smallest \u2018length\u2019 of paper that must be burned?\u00a0 This was an interesting problem, but how would the contestants express their solution?\u00a0 They would have to describe the motion of the cube as it passes through the plane.\u00a0 Some motions, even in two dimensions, are difficult to describe.\u00a0 But in three dimensions?\u00a0 We decided not to use this problem.<\/p>\n A third problem was about the \u201cDevil\u2019s Staircase,\u201d a now-classic way of using the Cantor set to define a step function which is continuous.\u00a0 It was decided that there is enough here to offer students who have not had an introduction to analysis.\u00a0 The analytic implications of the results could be appreciated as they learned more.<\/p>\n The reason for this seminar-on-wheels lay in the traditions of Soviet mathematics.\u00a0 In the USSR, teachers had very limited access to copying machines of any sort.\u00a0 Among other reasons, these could be used to reproduce unauthorized literature and so worked against control of information by the state.\u00a0 So test questions had to be written on the blackboard or even given orally.\u00a0 This led to traditions in testing\u2014and in contests\u2014which emphasized long-answer \u2018Olympiad\u2019 style problems, rather than the short answer problems more typical of American competitions.\u00a0 Many competitions included rounds that were conducted completely orally (for example, see Fomin and Kirichenko, 1994) .\u00a0 And the tradition of math battle or math wrangle (https:\/\/www.maa.org\/sites\/default\/files\/pdf\/sections\/math_wrangle.pdf<\/a>) also evolved partly from this circumstance.<\/p>\n 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.\u00a0 Later, at the camp itself, I witnessed the\u00a0 process of giving the students the problems, a process very different from any American contest I have known.\u00a0 The contestants gathered in a room, and the judges wrote the problem statements on a chalk board.\u00a0 They then explained the problems orally, taking questions from the audience to make sure the problem statements were clear.\u00a0 Finally the students were given three days to solve the problems and present them in math wrangle format.<\/p>\n 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.\u00a0 Competition is by \u2018town\u2019 (city).\u00a0 In the tradition of Russian\/Soviet competitions, problems all require solutions written out,\u00a0 and\u00a0 are selected so as to include both novice problem solvers and those with sophisticated background.<\/p>\n Konstantinov\u2019s work was central to numerous other initiatives.\u00a0 In 1978, he started the Lomonosov Tournament, a multi-subject competition named after Mikhail Lomonosov, the 18th<\/sup> 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.\u00a0 And well into his later years, Konstantinov continued working in Moscow High School 179, and helped to edit Kvant<\/em> magazine.\u00a0 \u00a0Matusov (2017) gives an account of his fresh approach to the classroom, as well as another set of personal reminiscences of Russian\/Soviet mathematical culture.<\/p>\n 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.\u00a0 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.\u00a0 You must choose the largest you can.\u00a0 After your choice, the integers stop coming.\u00a0 (This is a model for numerous life experiences\u2014even for high school students–from choosing a spouse or date to finding lodging along a highway.)\u00a0 In classic Russian style, Dynkin was able to break the problem down for his audience.\u00a0 I had seen this sort of exposition before, and was not surprised.\u00a0 What struck me, however, was the collegiality between Dynkin and Konstantinov.\u00a0 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.\u00a0 They were clearly members of the same community.\u00a0 It is now becoming more common to find such camaraderie in the American mathematical community.<\/p>\n 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.\u00a0 I recall him balancing on a beam which lay precariously across a swimming pool in Canada.\u00a0 He and I shopped for souvenirs in Australia, where he bought tiny koalas for each student in one of his classes.\u00a0 And in Amman, Jordan, we sat down to dinner at a conference we were both attending.\u00a0 The dinner plates were square.\u00a0 Konstantinov challenged me to find a reason for this shape.\u00a0 His reason?\u00a0 To make it easier to calculate their area.<\/p>\n Konstantinov\u2019s humor, his fresh attitude towards learning, his creative structuring of programs\u2026 they will all be missed, even as his legacy continues.<\/p>\n REFERENCES<\/p>\n Chan, Chi Ling (2015).\u00a0 Fallen Behind: Science, Technology, and Soviet Statism. Intersect, 8(3) (1-11). http:\/\/ojs.stanford.edu\/ojs\/index.php\/intersect\/article\/view\/691<\/a>.<\/p>\n Fomin, D., and Kirichenko, A.\u00a0 (1994) Leningrad Mathematical Olympiads 1987-1991<\/em>.\u00a0 Westford, MA: MathPro Press.<\/p>\n Gerovitch, S, (2013). Parallel Worlds: Formal Structure and Informal Mechanisms of Postwar Soviet Mathematics, Historia Scientiarum<\/em>, 22(3),181-200. https:\/\/www.academia.edu\/5366902<\/p>\n Kanel-Belov, Alexey et al.\u00a0 (2010) Interlocking of Convex Polyhedra: towards a Geometric Theory of Fragmented Solids<\/a>. Moscow Mathematical Journal., 10:2, 337–342, 2010 \u00a0(https:\/\/arxiv.org\/abs\/0812.5089<\/a> ).<\/p>\n 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<\/em>, Singapore: World Scientific Publishing Co. (43-86)<\/p>\n The Lomonosov Tournament, 1996, Math. Ed.<\/em>, 1997, Issue 1, 79\u2013106 (in Russian) http:\/\/www.mathnet.ru\/links\/075412a9f93379a3b310240dede3b677\/mo232.pdf<\/a><\/p>\n Matusov, Eugene (2017) Nikolai N. Konstantinov\u2019s Authorial Math Pedagogy for People with Wings, Journal of Russian & East European Psychology,<\/em> 54:1, 1-117, DOI: 10.1080\/10610405.2017.1352391\u00a0 http:\/\/dx.doi.org\/10.1080\/10610405.2017.1352391<\/a><\/p>\n Polyakova, T. (2010) “Mathematics Education in Russia before the 1917 Revolution”,\u00a0 in Karp, A., and Vogeli, B. (eds.) Russian Mathematics Education: History and World Significance<\/em>, Singapore: World Scientific Publishing Co. (1-42)<\/p>\n Saul, M. (1992). Love Among the Ruins. Focus, 12<\/em>(1), 1,6,7, https:\/\/www.maa.org\/sites\/default\/files\/pdf\/pubs\/focus\/past_issues\/FOCUS_12_1.pdf<\/a>. Accessed June 2020.<\/p>\n 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<\/em>, Singapore: World Scientific Publishing Co. (187-222)<\/p>\n <\/p>\n