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数学奥林匹克报数学竞赛问题讨论区我对竞赛有话说 → 陶哲轩 致中国的奥数爱好者

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陶哲轩 致中国的奥数爱好者
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陶哲轩 致中国的奥数爱好者
中国的奥数爱好者
        我很高兴自己有参加中学数学竞赛的经历(这个经历要追溯至20世纪80年代了!)与一群兴趣相同、水平相当的人一起竞赛,就像学校里任何其他的体育赛事一样,有一定的刺激。而且参加奥赛还可以有机会去国内外旅行,这种经历我想对所有的中学生强力推荐。
        数学竞赛还证明一点:数学并不只是分数和考试。但是数学竞赛同数学学习或者数学研究又迥然不同,比如说,你不要指望在研究生学习阶段所遇到的问题会像奥数问题一样单纯。  (尽管受过奥数训练的人也许能很快完成解题的一些步骤,但是大部分解题步骤可能还是得经过阅读文献、运用已知技巧、尝试例题或特例、寻找反例等这样更需要耐心、更冗长的过程。)
        此外,你在做奥数题时所学到的这种“传统”数学(例如,欧几里得几何、初等数论等等),看起来可能迥异于你在本科生和研究生阶段所学习的“现代”数学。虽然如果你稍稍深入下去,你会发觉传统数学其实依旧隐含于现代数学的基础之中。例如,欧氏几何学的传统定理提供了绝佳范例,开现代代数几何或微分几何之先河;同样,传统数论则为近世代数和数论之先端,等等诸如此类。所以,在学习现代数学时,你得做好准备要大大改变你的数学视角。  (但组合数学领域可能是一个例外,该领域还是有很多内容接近其传统根源,但这一块也在不断改变。)
        总而言之:要享受这些竞赛的乐趣,但不要忽视了你们数学教育中更为“枯燥”的内容,因为这些内容最终会显得更有用途。
Terence Tao   陶 哲 轩

James and Carol Collins Professor of Math
University of California Los Angeles
http://terrytao.wordpress.com/
    To Chinese International Math Olympiad Enthuasists:
I greatly enjoyed my experiences with high school mathematics competitions (all the way back in the 1980s!). Like any other school sporting event, there is a certain level of excitement in participating with peers with similar interests and talents in a competitive activity. At the olympiad levels, there is also the opportunity to travel nationally and internationally, which is an experience I strongly recommend for all high-school students.
Mathematics competitions also demonstrate that mathematics is not just about grades and exams. But mathematical competitions are very different activities from mathematical learning or mathematical research; don't expect the problems yon get in, say, graduate study, to have the same cut-and-dried, neat flavour that an Olympiad problem does.
(While individual steps in the solution might be able to be finished off quickly bysomeone with Olympiad training, the majority of the solution is likely to require instead the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and soforth.)
Also, the "classical" type of mathematics you learn while doing Olympiad problems (e.g. Euclidean geometry, elementary number theory, etc.) can seem dramatically different
from the "modern" mathematics you learn in undergraduate and graduate school, though if you dig a little deeper you will see tha t the classical is still hidden within the foundation of the modern. For instance, classical theorems in Euclidean geometry provide excellent examples to inform modern algebraic or differential geometry, while classical number theory similarly informs modernalgebra and number theory, and so forth. So be prepared for a significant change in mathematical perspective when one studies the modern aspects of the subject. (One exception to this is perhaps the field of combinatorics, which still has large areas which closely resemble its classical roots, though this is changing also.)
In summary: enjoy these competitions, but don't neglect the more "boring" aspects of your mathematical education, as those turn out to be ultimately more useful.
Terence Tao 陶哲轩
James and Carol Collins Professor of Math
University of California Los Angeles
http://terrytao.wordpress.com/
[此贴子已经被作者于2009/9/23 7:20:51编辑过]
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