I found it charming that the photo in that article showing Hannah displaying her proof on a screen the proof paper had flowers and trees drawn on it.
Hey, she was basically just a kid at the time. Just a very, very bright one!
No kidding; and I didn’t understand it any better when I was finished than when I started. This young woman has an astonishing intellect.
That’s why I said I found it charming, not stupid.
Doctorate programs are about mentorship and focusing on a very specific problem to advance knowledge in that niche. In that sense it makes perfect sense to do that. On a personal level I wonder if the breadth of undergrad electives and whatnot would be good to have. Just to be a well-rounded person, not that it would necessarily help w/ research (although new ideas can come from odd places).
I think you misunderstood me. I found it charming, too. The idea of “stupid” never even crossed my mind!
Yep. I wasn’t very good at math, either. The thing about calculus was that it gave me a glimpse – just a glimpse – into the wonders of mathematics. It seemed, for instance, marvelous and counter-intuitive that you could determine the precise area under a curve described by a known formula by breaking it up into a series of rectangles, and then computing the sum of the rectangles as their width approached zero and their number approached infinity, yielding a precise result at the limit. I still remember the picture in my first calculus textbook of Kresge Auditorium at MIT with its immense domed roof and rectangular window columns to illustrate the concept of integral calculus.
The mechanics of calculus (the ‘why’ of it) were never explained in any of the classes I took. I really feel like I missed out.
No worries.
I wondered that, too. It is very possible that she had a conditional admission, which allows her to do graduate work in math once she has taken (or while she is taking) some other courses. It would really depend on how they are thinking about this unusual admission, and I doubt if those details could be shared even if they were so inclined, due to student privacy issues.
I can understand the 6 universities that rejected her application to graduate school because of her lack of prerequisites – I don’t agree with it, but I can understand it as a matter of policy. But what really pisses me off are the two that admitted her, and the admission was then overturned by administration bureaucrats. That’s some real bureaucratic bullshit there. Fortunately, she was offered admission at two others with more enlightened administrations.
Same here. I took two semesters of calculus in college, but both classes were “business calculus,” specifically designed for business majors. The classes taught how to use derivatives, integrals, etc.*, but compared to the “real” calculus classes (which my friends who were engineering majors took), the business-calc classes didn’t explain the why behind those equations.
*- my understanding is that those calculus tools are mostly relevant for business specializations like finance and economics, as they underlie the models used.
Yeah. That pissed me off too. My brief foray into post-graduate education almost 40 years ago taught me all that I needed to know about university bureaucracy.
You have to [NB it is never too late!!] take the (pure) mathematics classes, not engineering courses. Also, perhaps some places sweep some of the important foundational material into “honors” classes or a subsequent “real analysis” class instead of first-semester calculus where it belongs.
I guarantee she had something like that to deal with; she would not be the first. Fortunately, the dean and so on have some power to bend the rules, conditionally [on taking certain classes] and otherwise.
Beyond that, I find it distasteful, to say the least, to suggest that this student did something she should not have, unless whoever is casting aspersions knows something we do not.
[PS well, she might have avoided talking to journalists and posing for photos. Then there would not have been this discussion of her university admissions on some random, not even mathematical, online message board. Could have been a calculated decision, however— I know a couple of people who are first on the list whenever the New York Times needs a quote about anything involving mathematics or theoretical physics.]
I was a physics undergrad, and had an 8AM calculus class my first semester, where I earned a D. I didn’t much understand calculus until upper level physics + statistics.
A fair number of non-US masters candidates in our physics program had to drop down and take some undergraduate classes to fill requirements that they either didn’t have, or which weren’t sufficient, in their undergrad degree programs. I can certainly see doing something like that for her if something’s missing.
Because a PhD isn’t just a credential to show how much stuff you’ve learned, and it doesn’t involve just learning more things. It involves working with other people to develop new knowledge, and teaching others to do the same. Though technically you’re a student, it’s an actual job.
No offense, but if you don’t get it, you don’t get it, and it sounds like you’re unlikely to get it.
You might join a particular PhD program for the opportunity to learn from and work with a particular person or group. There are also advanced classes you can take, seminars, colloquia, workshops, etc., that get you used to a career in mathematical research (and if you are that brilliant, people will know it and remember you). It’s not too complicated to understand, all the legitimate criticism of modern academia notwithstanding.
Mine was definitely geared to engineers. What’s weird is that while I barely squeaked by three semesters of calculus, I aced differential equations.
Where’s that quote from?
That’s a pretty heavy emphasis in AP Calculus. I can agree with that: For most of the things you use calculus for, you need to understand what calculus really means, in order to set up your integrals, but once you’ve set up an integral, you can just plug it into Maple or Mathematica or something. Knowing the techniques of doing the integrals is, of course, useful, but it’s not by far the most important part.
I was an astronomy major in undergrad, and then a physics major in grad school. This gave me an extremely strong background in some areas of physics, but almost none at all in others, and as a result I had to take undergrad courses in E&M and quantum mechanics as a graduate student (followed eventually, of course, by graduate-level classes in those same subjects).
The the standard pedagogy for integral calculus, which has certainly has applications but is presented up front in its theoretical basis of taking the limit of a function as it goes to a point (which is not very intuitive) and then shifts into the mechanics of manipulating equations such that they can be integrated, so you spend very little time doing anything useful with them. This makes sense from a fundamentals of mathematics point of view, but like studying set theory before going into algebraic manipulation, it is basically creating a foundation that has little reference for the user until they actually get into more advanced mathematics. In contrast, differential equations is highly applied and typically taught using physically intuitive examples like water flow into and out of a tank. While actually solving more complicated (partial) differential equations can be quite challenging.
Personally, I learned both algebra and calculus from texts I picked up at a used bookstore or swap meet, both of which apparently had non-standard approaches, and so I learned them in the ‘wrong’ way so even though I could ‘do the work’ I ran into a lot of problems doing it correctly per the standard pedagogy. (I still have flashbacks to my algebra teacher standing over my desk and screaming at me to “foil it, foil it, FOIL IT!” in front of the entire class, I guess to shame me into solving quadratic equations in the weird, rote fashion she learned that took up an entire page rather than just solving it in my head using the quadratic formula.) I struggled a bit with the first couple of semesters of calculus but stepped on it when we got to vector calculus and then ordinary and partial differential equations. I was then forced to take two more classes that were basically more vector (and supposedly tensor, albeit with minimal coverage) calculus and a little bit of spherical trigonometry and linear algebra, which gave me the credits for a math minor but were mostly just retreads of stuff I already understood from previous math and other classes.
Academia is as much about networking, access to resources like journals and computing licenses that are prohibitively expensive to working on your own, and the opportunity to be exposed to new ideas and methods both within and outside of your field. For a theoretical mathematician interested in the emerging frontiers of mathematics, getting a PhD and working in academia is the industry. Although it makes for an amusing scene, the National Security Agency recruiting a brilliant but undisciplined and uncredentialed math genius is one of the most implausible elements in the film. They (and industries employing mathematicians such as software companies making analytical tools or working on advanced ‘deep learning’ AI systems) are looking for people with degrees and a background of working in applied mathematics in the specific area of interest, not someone wandering through unresolved conjunctures.
Stranger
Right, exactly. My comments were more to the confused comments who seemed to think a person could invent whole new fields of mathematics just by thinking really hard, just because they skipped HS and college. Brainpower is crucial, but at some point you need more grist for the mill, food for thought, people to bounce things off of, and a sense for which problems are most valuable to solve. You can’t get that working in a silo.
And most crucially, if you’re that bright, you’ll probably develop enough knowledge that you want to teach and publish. That’s not a skill that comes on its own, and it’s not an opportunity that falls out of a tree. A PhD program offers experience and reach.