Great Mathematicians on Precocity, Speed and Math Competitions

post by multifoliaterose · 2010-10-11T10:45:52.139Z · LW · GW · Legacy · 0 comments

In a 2003 interview, Terence Tao answered the question

What advice would you give to young people starting out in math (i.e. high school students and young researchers)?

by saying

Well, I guess they should be warned that their impressions of what professional mathematics is may be quite different from the reality. In elementary school I had the vague idea that professional mathematicians spent their time computing digits of pi, for instance, or perhaps devising and then solving Math Olympiad style problems.

In The Case against the Mathematical Tripos mathematician GH Hardy wrote

It has often been said that Tripos mathematics was a collection of elaborate futilities, and the accusation is broadly true. My own opinion is that this is the inevitable result, in a mathematical examination, of high standards and traditions. The examiner is not allowed to content himself with testing the competence and the knowledge of the candidates; his instructions are to provide a test of more than that, of initiative, imagination, and even of some sort of originality. And as there is only one test of originality in mathematics, namely the accomplishment of original work, and as it is useless to ask a youth of twenty-two to perform original research under examination conditions, the examination necessarily degenerates into a kind of game, and instruction for it into initiation into a series of stunts and tricks.

In The Map of My Life mathematician Goro Shimura wrote of his experience teaching at a cram school

I discovered that many of the exam problems were artificial and required some clever tricks. I avoided such types, and chose more straightforward problems, which one could solve with standard techniques and basic knowledge. There is a competition called the Mathematical Olympic, in which a competitor is asked to solve some problems, which are difficult and of the type I avoided. Though such a competition may have its raison d’ˆetre, I think those younger people who are seriously interested in mathematics will lose nothing by ignoring it.

In his lecture at the 2001 International Mathematics Olympiad, Andrew Wiles said

Let me then welcome you not only to this event but also to the greater world of mathematics in what many of us believe is now a golden age. However let me also warn you — whatever the route you have taken so far, the real challenges of mathematics are still before you. I hope to give you a glimpse of this. What then distinguishes the mathematics we professional mathematicians do from the mathematical problems you have faced in the last week? The two principal differences I believe are of scale and novelty. First of scale: in a mathematics contest such as the one you have just entered, you are competing against time and against each other. While there have been periods, notably in the thirteenth, fourteenth and fifteenth centuries when mathematicians would engage in timed duels with each other, nowadays this is not the custom. In fact time is very much on your side. However the transition from a sprint to a marathon requires a new kind of stamina and a profoundly different test of character. We admire someone who can win a gold medal in five successive Olympics games not so much for the raw talent as for the strength of will and determination to pursue a goal over such a sustained period of time. Real mathematical theorems will require the same stamina whether you measure the effort in months or in years [...] The second principal difference is one of novelty [...] Let me stress that creating new mathematics is a quite different occupation from solving problems in a contest. Why is this? Because you don't know for sure what you are trying to prove or indeed whether it is true.

In his Mathematical Education essay, Fields Medalist William Thurston said

Related to precociousness is the popular tendency to think of mathematics as a race or as an athletic competition. There are widespread high school math leagues: teams from regional high schools meet periodically and are given several problems, with an hour or so to solve them.

There are also state, national and international competitions. These competitions are fun, interesting, and educationally effective for the people who are successful in them. But they also have a downside. The competitions reinforce the notion that either you ‘have good math genes’, or you do not. They put an emphasis on being quick, at the expense of being deep and thoughtful. They emphasize questions which are puzzles with some hidden trick, rather than more realistic problems where a systematic and persistent approach is important. This discourages many people who are not as quick or as practiced, but might be good at working through problems when they have the time to think through them. Some of the best performers on the contests do become good mathematicians, but there are also many top mathematicians who were not so good on contest math.

Quickness is helpful in mathematics, but it is only one of the qualities which is helpful. For people who do not become mathematicians, the skills of contest math are probably even less relevant. These contests are a bit like spelling bees. There is some connection between good spelling and good writing, but the winner of the state spelling bee does not necessarily have the talent to become a good writer, and some fine writers are not good spellers. If there was a popular confusion between good spelling and good writing, many potential writers would be unnecessarily discouraged.

In Recoltes et Semailles, Fields Medalist Alexander Grothendieck wrote

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

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