Mathematics at MIT, Harvard, Princeton

There is interesting data to show that MIT takes math students more seriously than Harvard and Princeton. By Michael Sipser’s suggestion I looked at the Putnam Competition results. Out of the top 74 scorers of 2007, 21 were from MIT, 9 from Harvard and 7 from Princeton. Keep in mind that the total freshman enrollment at MIT is much lower than at Harvard or Princeton. This story repeated itself in 2008: out of top 79 scorers 23 were from MIT, 11 from Harvard and 11 from Princeton.

Ironically, MIT’s team didn’t win Putnam in those years. MIT’s team won the third place after Harvard and Princeton. If you look at the results more closely, you will notice that had MIT arranged teams differently, MIT would have won.

It appears that MIT put their three top scorers from the previous year on their lead team. MIT shouldn’t assume that those three continue to be their strongest competitors. Instead they should probably test their students right before the Putnam competition, because if you look at MIT’s top individual performers, had they been on a team together, they would have won.

Maybe MIT should rethink its algorithm for creating teams, or maybe we should just wait. As it is obvious that MIT is more serious about math, all top math students may want to go to MIT in coming years. If this happens, the mathematics field will be absolutely dominated by MIT.

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7 Comments

  1. Felipe Pait:

    “total freshman enrollment at MIT is much lower than at Harvard or Princeton” was an opportunity to text Wolfram Alpha. But Math enrollment at MIT is probably larger, no? The shame for me is how few Yalies show up….

  2. JBL:

    On the AoPS forum (where you posted a link to this discussion), Kent Merryfield has engaged in some more detailed analysis of winning teams (I don’t have a precise link handy). If you allow MIT to rearrange its team after the competition each year, you surely must allow every other school to do so as well, and once you do this it is not necessarily clear that MIT would have won in the years you mention (due primarily to the strange algorithm the Putnam uses to determine team ranks and the fact that they don’t release the actual places of the top students), though I think they wouldn’t have placed lower than 2nd.

    I know that Harvard also follows a “best three from the previous year” strategy, and it seems to work well for them. This suggests that the algorithm itself is probably not the problem.

    I also don’t think your conclusion follows from this analysis (even excepting the conflation of “having good results in Putnam” with “being serious about math”) — MIT’s undergraduate classes are certainly smaller than Harvard’s or Princeton’s (though “much lower” seems like an exaggeration in the case of Princeton — the difference is about 700 students out of more than 4000), but MIT has vastly more students who take nontrivial amounts of mathematics than either Harvard or Princeton (if for no other reason than that 18.01 and 02 are required at MIT). A much more sensible comparison might be based on the number of students who take an abstract algebra class, or even just who sign up to take the Putnam. (This latter number is in the high double-digits or low triple-digits at Harvard, I believe — I have no idea what it might be for MIT or Princeton.)

    For what it’s worth, when I was near the end of high school, the top students on the New York City Math Team seemed to favor Harvard over MIT (though not by a huge amount). In recent years there has been a noticable shift in the direction of MIT. I think, however, that there are more than enough talented math students around the country distributing themselves widely enough to keep the Putnam exam competitive in coming years.

  3. Tanya Khovanova:

    JBL,

    I agree that my enrollment size argument is not clean. Also, it goes without saying that if I allow MIT to rearrange its team, then every school must be allowed to do that.

    On the other hand, it could be that at MIT people tend to become more interested in research and less interested in competitions. This can explain why previous Putnam fellows no longer do well. If this tendency is more pronounced at MIT than at Harvard, then it could also explain why the algorithm works for Harvard and doesn’t work for MIT.

  4. JBL:

    Hi Tanya,

    Thanks for your response (and e-mail). I thought AoPS user “blahblahblah” had a very insightful comment, which I paraphrase/expand on here: presumably every student has some mean expected number of points and some distribution around that mean. When we talk about the “Putnam strength” of an individual, we probably mean this expected value, but we only get to choose a single realization of her score from this distribution. Because MIT has so many students taking the exam with very high expected scores, the likelyhood that (on any given realization) the student with the highest score is not actually the student with the highest expected score is quite high (in a way that is less true of some school where the Putnam strength drops off more quickly). This means that MIT will frequently not choose its top three students (even if it always chose the students with the highest expected scores), but also that the algorithm “use the top 3 from the previous administration” will almost always fail to choose the students with the highest expected scores.

  5. Anonymous:

    One possible comparison is number of science or engineering majors. This isn’t perfect, but the numbers are available on the web.

    In 2008, 459 Harvard students graduated with degrees in science or engineering. 217 Princeton students graduated with degrees in science and 177 in engineering. This gives a total of 853.

    For MIT I can’t find these statistics, but the 2008-9 enrollment statistics on the “MIT facts” page show that about 87% of the students are majoring in science or engineering. Assuming 1000 students graduating per year (there are 4153 undergraduates currently enrolled, but let’s assume some fail to graduate), that should be about 870 science or engineering majors per year.

    By this measure, you should expect MIT to have slightly more high-scoring students on the Putnam exam than Harvard and Princeton combined.

  6. ryrma:

    MIT statistics:
    http://web.mit.edu/registrar/www/stats/ (not for 2008-09, though)

    968 students graduated from School of Engineering and School of Science in 06-07 (out of 1167 total) http://web.mit.edu/registrar/www/stats/deg0607.html

    So I guess the argument can be made that H and P students actually fire much better than could be expected from sheer numbers…

  7. Kitchener L. Harding:

    The Riemann Hypothesis was solved and posted at http://www.slopedope.com today 15 Dec 2011. It shows that the Menorah Number Theory has a Fractal at its center that shows zero is infinite with prime numbers that are expandable and has unique patterns. My 16 year old son helped me post it online. He is just one of the College by 12 family. Starting college by 12 is standard practice for many. Why wait for an Ivy league invitation, when they don’t have the good sense to look around at the real high achievers?

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