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<p>But that 0.001% of the smartest students in the world may represent 20-30% of a class in mathematics at Harvard or MIT. Over 40% of ALL top 100 scorers at the Putnam math competition are undergrads at MIT and another 20% at Harvard, 10% at Princeton and Caltech. That is an astounding 80% of the top 100 from just four schools. Many of these students directly take advanced undergrad level or graduate level classes as freshmen. A number of students are recruited from the top talent in China, India, Russia through the IMO. There is immense pressure to recruit world class faculty to teach these 16-20 year old geniuses. It is not different from Duke or UCONN seeking the best basketball coaches for their teams. For MIT and Harvard’s math departments (and beyond), success is often defined by how well they do at the Putnam. I recall sitting in on one of my D’s math classes freshman year. This was just multi-variable calculus, the second easiest math class at MIT. You would think they would bring out some junior faculty to teach the class. Not at all! The class was taught by a young brilliant French mathematician, not even 30 years old, former IMO gold medalist himself, who was the youngest math professor ever granted tenure at MIT at the age of 28. He also happened to teach one of the most advanced graduate level math seminars. His pace was so fast, his proofs so detailed and mastery of the subject so convincing, he would routinely get a standing ovation from the students after each class! He would cover points well beyond the core material so that even the most brilliant of his students would be stretched to their limits! And this was a very difficult audience to please. This happens not just at MIT. Harvard has a notoriously difficult undergraduate math class, Math 55, arguably one of the hardest freshman math classes in the US also taught by the best math faculty Harvard has to offer. </p>
<p>At the very top institutions, the distinction between undergrad and grad students becomes blurred as a number of undergrads routinely take grad level courses. The depth and breadth of the courses offered is such that you can’t afford to put “average” faculty to teach any of the courses, even at the freshman level.</p>