In this Blog, the invited presentation of Siu Man Keung at the IMO Forum during IMO 2016 in Hong Kong, is reproduced here with his permission. We trust readers will find his title and presentation provocative, and the three problems in the APPENDIX stimulating. (The appendix is available for downloading separately at the provided link).
Does society need IMO Medalists?
(Invited talk at the IMO Forum at the Hong Kong Polytechnic University on 11 July 2016)
Siu Man Keung
Department of Mathematics
University of Hong Kong
The title of this talk that sounds provocative is not chosen with any intention to embarrass the organizers and participants of the event of IMO (International Mathematical Olympiad). It should be seen as the sharing of some thoughts on this activity, or more generally on mathematical competitions, by a teacher of mathematics who had once helped in the coaching of the first Hong Kong Team to take part in the 29th IMO held in Canberra in 1988 and in the coordination work of the 35th IMO held in Hong Kong in 1994. The speaker tries to look at the issue in its educational context and more broadly in its socio-cultural context.
About the speaker
SIU Man Keung obtained his BSc from the Hong Kong University and went on to earn a PhD in mathematics from Columbia University. Like the Oxford cleric in Chaucer’s The Canterbury Tales, “and gladly would he learn, and gladly teach” for more than three decades until he retired in 2005, and is still enjoying himself in doing that after retirement. He has published some research papers in mathematics and computer science, some more papers of a general nature in history of mathematics and mathematics education, and several books in popularizing mathematics. In particular he is most interested in integrating history of mathematics with the teaching and learning of mathematics and has been participating actively in an international community of History and Pedagogy of Mathematics since the mid-1980s. He has devoted much of his time in offering a course titled Mathematics: A Cultural Heritage in the tradition of liberal studies for undergraduates from various Faculties of the Hong Kong University for a decade during the 2000s as well.
Does society need IMO Medalists?
Does society need IMO Medalists? No, society does not “need” IMO Medalists. Society does not even “need” mathematicians. Does this mean my 30-minute talk will end here? Stopping here and now would amount to an admission of wrong choice of my profession in all these years, so I should go on talking. You will notice that I put the word “need” in quotation marks. Society does not “need” (in quotation marks!) IMO Medalists or mathematicians, but society needs (no quotation mark!) MTR maintenance workers, garbage collectors, street cleaners, plumbers, electricians, etc. Now, perhaps you know what I mean. Let us get back to IMO.
After 22 years IMO comes back to Hong Kong as the host. Hong Kong hosted the 35th IMO in the summer of 1994. I like to mention two Medalists in that particular IMO. One is Maryam Mirzakhani of Iran, who became the first female mathematician to receive a Fields Medal at the International Congress of Mathematicians in 2014. The other one is Subash Ajit Khot of India, who was awarded the Nevanlinna Prize at the same Congress.
In a talk I gave in 2012 with the title “The good, the bad and the pleasure (not pressure) of mathematics competitions” I outlined certain good and bad points of mathematics competitions. Allow me to repeat them here in summary. [For a more detailed discussion, see: M.K. Siu, Some reflections of a coordinator on the IMO, Mathematics Competitions, 8 (1) (1995), 73-77; M.K. Siu, The good, the bad and the pleasure (not pressure!) of mathematics competitions, Mathematics Competitions, 26(1) (2013), 41-58.]
Good points: Nurturing of (1) clear and logical presentation, (2) tenacity and assiduity, (3) “academic sincerity”; moreover, arouse a passion for and pique the interest in mathematics.
Bad points: (1) competition problems versus research, (2) over-training?
We further ask: Is the passion for the subject of mathematics itself genuine? Can the interest be sustained?
Let me further explain the point about competition problems versus research through three examples (see APPENDIX available as a PDF at: http://dynamicmathematicslearning.com/appendix-MKSiu-IMO2016.pdf ).
What do we see from these three examples? It makes me think that there are two approaches in doing mathematics. To give a military analogue one is like positional warfare and the other guerrilla warfare. The first approach, which has been going on in the classrooms of most schools and universities, is to present the subject in a systematically organized and carefully designed format supplemented with exercises and problems. The other approach, which goes on more predominantly in the training for mathematics competitions, is to confront students with various kinds of problems and train them to look for points of attack, thereby accumulating a host of tricks and strategies.
Each approach has its separate merit and they supplement and complement each other. Each approach calls for day-to-day preparation and solid basic knowledge. Just as in positional warfare flexibility and spontaneity are called for, while in guerrilla warfare careful prior preparation and groundwork are needed, in the teaching and learning of mathematics we should not just teach tricks and strategies to solve special type of problems or just spend time on explaining the general theory and working on problems that are amenable to routine means. We should let the two approaches supplement and complement each other in our classrooms. In the biography of the famous Chinese general and national hero of the Southern Song Dynasty, Yue Fei (1103-1142) we find the description: “(Setting up the battle formation is the routine of the art of war. Maneuvering the battle formation skillfully rests solely with the mind.)”
Sometimes the first approach may look quite plain and dull, compared with the excitement acquired from solving competition problems by the second approach. However, we should not overlook the significance of this seemingly bland approach, which can cover more general situations and turns out to be much more powerful than an ad hoc method which, slick as it is, solves only a special case. Of course, it is true that frequently a clever ad hoc method can develop into a powerful general method or can become a part of a larger picture. A classic case in point is the development of calculus in history. In ancient time, only masters in mathematics could calculate the area and volume of certain geometric objects, to name just a couple of them, Archimedes (c. 287 B.C.E. – c. 212 B.C. E.) and LIU Hui ( 3rd century). In hindsight their formula for the area of a circle, A = (1/2) x C x r , embodies the essence of the Fundamental Theorem of Calculus. With the development of calculus since the seventeenth century and the eighteenth century, today even an average school pupil who has learnt calculus will be able to handle what only great mathematicians of the past could have resolved.
Since many mathematics competitions aim at testing the contestants’ ability in problem solving rather than their acquaintance with specific subject content knowledge, the problems are set in some general areas which can be made comprehensible to youngsters of that age group, independent of different school syllabi in different countries and regions. That would cover topics in elementary number theory, algebra, combinatorics, sequences, inequalities, functional equations, plane and solid geometry and the like. Gradually the term “Olympiad mathematics” is coined to refer to this conglomeration of topics. One question that I usually ponder over is this: why can’t this type of so-called “Olympiad mathematics” be made good use of in the school classroom as well? If one aim of mathematics education is to let students know what the subject is about and to arouse their interest in it, then interesting non-routine problems should be able to play their part well when used to supplement the day-to-day teaching and learning.
Let us get back to the question in the title of the talk: Does society need IMO Medalists? No, society does not “need” IMO Medalists. Society does not even “need” mathematicians. But society needs “friends of mathematics”. A “friend of mathematics” may not know a lot of mathematics but would understand well what mathematics is about and appreciate well the role of mathematics in the modern world.
The mathematician Paul Halmos once said, “It saddens me that educated people don’t even know that my subject exists.” Allen Hammond, editor of Science, once described mathematics as “the invisible culture”. On the other hand, perhaps it is a blessing to remain not that visible! Two months ago I read in the news (Associated Press, May 7, 2016): “Ivy League Professor Doing Math Equation on Flight Mistaken for Terrorist”. An American Airline passenger seated next to Guido Menzio of UPenn suspected the unfamiliar writings of the professor were a code for a bomb. It led to Professor Menzio being taken away from the plane to be interrogated!
In ancient China the third-century mathematician LIU Hui said, “ (The subject [mathematics] is not particularly difficult by using methods transmitted from generation to generation, like the compasses [gui] and gnomon [ju] in measurement, which are comprehensible to most people. However, nowadays enthusiasts for mathematics are few, and many scholars, much erudite as they are, are not necessarily cognizant of the subject.)”
Why is it like that?
Here is a passage taken from a book: “Central to my argument is the idea that ***** distinguished by a self-conscious attention to its own ***** language. Its claim to function as art derives from its peculiar concern with its own materials and their formal patterning, aside from any considerations about its audience or its social use.” Can you guess what the missing words are?
This passage is taken from the book by Julian Johnson, Who Needs Classical Music? Cultural Choice and Musical Value (2002). The missing words are “classical music” and “musical”. However, the passage would ring equally true if “classical music” is replaced by “mathematics”!
In the same book the author says, “… that it [meaning classical music] relates to the immediacy of everyday life but not immediately. That is to say, it takes aspects of our immediate experience and reworks them, reflecting them back in altered form. In this way, it creates for itself a distance from the everyday while preserving a relation to it.” Mathematics is also like that. This explains why it is not easy to bring mathematics to the general public. To become a “friend of mathematics” one needs to be brought up from school days onward in an environment where mathematics is not only enjoyable but also makes good sense. In the preface to a textbook [Alice in Numberland: A Students’ Guide to the Enjoyment of Higher Mathematics (1988)] the authors, John Baylis and Rod Haggarty, remark, “The professional mathematician will be familiar with the idea that entertainment and serious intent are not incompatible: the problem for us is to ensure that our readers will enjoy the entertainment but not miss the mathematical point, […]”
My good friend, Tony Gardiner, an experienced four-time UK IMO team leader, once commented that I should not blame the negative aspects of mathematics competitions on the competition itself. He went on to enlighten me on one point, namely, a mathematics competition should be seen as just the tip of a very large, more interesting, iceberg, for it should provide an incentive for each country to establish a pyramid of activities for masses of interested students. It would be to the benefit of all to think about what other activities besides mathematics competitions can be organized to go along with it. These may include the setting up of a mathematics club or publishing a magazine to let interested youngsters share their enthusiasm and their ideas, organizing a problem session, holding contests in doing projects at various levels and to various depth, writing book reports and essays, producing cartoons, videos, softwares, toys, games, puzzles, … .
Finally the question boils down to one in an even more general context: Does society need ME? We frequently hear about the cliché “No one is indispensable!” But please bear in mind that everyone has his or her worth and can do his or her part to make this world a better place to live in. An IMO medalist is no exception!