A platform for problem-solving

Shailesh Shirali

Mathematical Olympiads have been taking place in the country for some decades now, and it is relevant to examine the significance they hold for the country in a wider sense.

Mathematical Olympiads originated in the east European countries. In 1894, a competition known as the Eötvös began in Hungary; it quickly became prestigious; many famous mathematicians have in one way or the other been associated with it. Russia, Romania, and Bulgaria soon started similar events of their own. A particularly notable event is the town-based Tournament of the Towns; it originated in Russia and continues to be held in parts of the world, in modified forms. The event known as the International Mathematical Olympiad (IMO) started in 1959 in Romania. For two decades, participation was limited to the Eastern Bloc countries, but from the mid-1970s, increasingly many countries started to take part, and the latest count stands at over 100.

The distinguishing feature of these events is the quality of the problems posed in the examination. They are original, in the sense that they are not simply known results expressed in a different way. To solve them, one typically requires original and critical thinking of a high order. The time given to solve the paper is indicative of this: in the IMO, six questions are given for solution over two sessions: one session per day, 4.5 hours per session! Some IMO questions have gone on to become classics in the field, offering starting points for collaborative research and intense creative inquiry. I can testify to this from my own personal experience.

IMO-logo In India, the mathematical Olympiads are organized by the National Board for Higher Mathematics. The national level event is the Indian National Mathematics Olympiad (INMO); it has been held every year since 1989. It is preceded by various local Olympiads (the Regional Mathematical Olympiads, held in individual states) in a pyramidal structure. Students who qualify for the INMO are invited for an IMO training camp held every year in May-June at the Homi Bhabha Centre for Science Education in Mumbai, and at the end of this camp, a six-member team is selected to represent India in the IMO, which is held in July in different countries. The event was last hosted by India in 1996. In 2015, the event was hosted by Thailand.

Among the many charming traditions of the IMO is the procedure for the selection of the contest problems: problems are invited from the participating countries themselves, and from the problems received, 30 are shortlisted prior to the event. The selection of the final six problems from this list is done by the entire group of team leaders of the participating teams (the team leader is typically a senior academic who travels to the venue of the contest two days in advance of the rest of the team). Over several hectic sessions during these two days, and a great deal of discussion which sometimes can get extremely animated, the selection is done and the problems reworded in a suitable manner. As noted earlier, in the process, several problems of rare beauty and elegance emerge.

For historical reasons, the topics on which the problems are based belong to what is technically known as ‘elementary mathematics’ – high school algebra, high school geometry, number theory and combinatorics. Note that calculus is excluded from this list. (Comment: The word ‘elementary’ does not mean ‘easy’! Elementary mathematics is that zone of mathematics in which we do not consider infinite processes.)

It should not come as a surprise that many great mathematicians of current times have begun their careers via the IMO. Here are some particularly well-known names: Terence Tao, a prodigy from Australia; Grigori Perelman of Russia, who proved a conjecture that they had been unsolved for a century; Maryam Mirzakhani from Iran, who is the first woman mathematician to be awarded the prestigious Fields medal. In India too, several mathematicians who are well-known on the world stage have come through the IMO: Sucharit Sarkar, K Soundarajan, Subhash Khot.

These days, there are numerous other events of a similar nature held across the world, particularly Europe and North America, but they are not as well-known as the IMO, which represents the high point of all such events. At the undergraduate level there are in comparison very few such events. The best known among these is the Putnam Lowell competition, held in USA. In India a similar competition is held, but it remains comparatively unknown. (It is organized by Bhaskaracharya Pratishthana of Pune.)

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Regarding the Olympiad activities held in India, what has been described above is the centralized, official one; but in addition to this, there are a number of lesser-known Olympiads. Many of these are run by private agencies and are, regrettably, of very poor quality; possibly some of these are simply money-making devices which cash in on the fact that Olympiads have acquired a certain elite connotation in recent years, and schools across the country would like their children to participate in them and possibly win some prizes; likewise the parents. But a good many of these competitions are nothing more than quizzes of a routine nature, with negligible educational value.

I come now to another aspect of the Olympiads which is relevant for us in India: the competitive aspect. Let me pose two questions: (a) Why do Olympiads of this nature exist at all? (b) Why do students take part in them? The two questions, though linked, need to be answered separately.

As regards (a), I feel that the Olympiads offer a glorious opportunity for exposure to non-routine problems and in general to the beauty in mathematics. It is a sad but readily verified fact that the mathematics curricula in India are extremely content heavy, emphasizing formulas which typically need to be memorized; they offer very limited scope for exploration. There is also little or no exposure to the culture and beauty of mathematics, in the sense of collaborative exploration of open problems, and to the culture of reading good books in mathematics. A comment made by Manjul Bhargava, the Canadian mathematician of Indian origin who was awarded the Fields Medal in 2014, is of relevance here: he described mathematics teaching in India as “robotic”. Seen from this perspective, the Olympiads offer us a way of correcting the balance, and they can do so effectively and beautifully.

What about (b)? There are certainly many students who take part in the Olympiads for reasons related to the above; perhaps some of them end up as mathematicians or scientists in some other field. But many students take part for reasons that are less clear. They are probably pressurized to do so by well-meaning adults. Ours has become a highly competition and examination driven society, especially during the last two to three decades, with exams such as the JEE dominating our mindsets. What a to-do we make of students who ‘top’ the JEE (or for that matter, the board examinations), how we worship the gold medallists! It is sad when we behave like this. What we see here is the flip side of the Olympiad movement.

I may add here that a good many countries take the competitive aspect of the IMO very seriously; such countries may invest a great deal of human resource into their IMO training programs and go on to do extremely well in the IMOs. To me, it is a matter of great regret to see nationalism take root in the culture of mathematics.

Another factor to be pondered is that owing to the poverty of exposure at the school level to non-routine problems, most students who take part in the Olympiads are hugely unprepared for them; they simply cannot tackle the kinds of problems they encounter in these contests. In consequence they do very poorly, and this can act as a substantial discouragement to them, especially if they regarded themselves as reasonably good in the subject prior to the contest.

Olympiads in general offer an excellent educational resource, but we need to use them wisely.

The author is Principal of Sahyadri School KFI, Pune. He has been in the field of mathematics education for three decades, and has been closely involved with the Math Olympiad movement in India. He is the author of many mathematics books addressed to high school students, and serves as an editor for At Right Angles. He can be reached at shailesh.shirali@gmail.com.

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