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When I was trying to choose a PhD supervisor one of the things I did was read through recent grads' acknowledgments. While no one ever mentioned their advisor with anything but gracious words, you could get a pretty good idea of what working with that faculty member was like.
It's true that professors rely on grad students/postdocs to do work. And, at least in my experience, advisors are actually pretty good about giving students credit for their work -- having successful students reflects well on the advisor. But a student is even more dependent on their advisor than vice versa. It's not like undergrad where the main thing that matters to your future employment is to collect the diploma, because for most fields the main reason to get a PhD is to continue in academia, and academia right now is an extreme employer's market. The things you need to leave grad school with are 1) impressive recommendation letters, like "best student in N years, reminiscent of <mid-career hotshot> at that age", and 2) (lots of) refereed publications. If you manage that, the diploma should be automatic.
Yes, you can push back against advisors who require 12 hrs a day in lab. But if that means you take longer to produce work, your letters might be just good instead of positively glowing, which might mean you fail to launch in academia. Several hundred other people will apply for each tenure track job you apply to; those with "just good" letters tend to get crowded out. The tenured advisor might have a bruised ego because their publication rate has slowed, or be more frazzled because they have to save money and write more proposals, but at least they still have a job.
Also, PIs themselves generally work a lot as well (often the ones insisting on lots of hours from their students think, rightly or wrongly but based on their own experience, that that's the only way to succeed....). I agree academia is broken, but think it's at a deep structural level, and more complicated than schools exploiting students and hanging them out to dry.
Just having the good new ideas isn't really enough, though. You have to be really persistent about figuring out all the details and making them work. This is related to, but definitely not the same as, being fascinated/obsessed by the topic.
Of course Einstein had great ideas. But he also spent many years working out the consequences of, eg, his first ideas about the fixed speed of light in vacuum and its consequences in physics, initially during downtime at his patent office job. Nearly all of the impact of the theory is in that working-out.
So I'm totally on board with everyone knowing something about the liberal arts and trying to expand your world outside your traditional scope.
But I think "understanding" can mean just making connections within a single tech field, even without involving liberal arts. For example, a grade school math problem: "Assume the earth is a perfect sphere with radius 6378 km, and you have a piece of string just long enough to reach all the way around the earth's equator at the earth's surface. How much longer would your string have to be to make a perfect circle exactly one meter above the earth's equator at every point?"
The answer is 2(pi) meters. That's true for any spherical planet of any size -- that's what it means for the derivative of 2(pi)(r) with respect to r to be 2(pi). That is sometimes not the first thing people think of though, because of the grade-school context they associate with this problem....
Nice article! I guess it's obvious in retrospect, but I hadn't known of all the systematized study devoted to this topic. I'm happy to learn about it because I've found myself thinking about effective teaching and learning pretty often (I'm an academic), and what to do about "the stuff where, when you try to explain it concretely to someone else, your explanation doesn't really make sense unless the other person already knows what you're talking about".
In subjects I've tried to learn and teach, my experience is that talking to someone with a lot of such knowledge really only gives you an idea of the sub-topics and considerations you should try to understand better on your own. It is helpful in narrowing down what you should prioritize and maybe giving you a useful point of view to organize your thoughts from, but that doesn't save you from doing the thinking and understanding for yourself.
I agree that emulation helps somewhat by forcing you to make choices that are reasonable even if, as a beginner, you lack the knowledge to choose wisely yourself. But if the ultimate goal is to come up with new ideas using the knowledge, I think there's such a thing as too much emulation. You don't want to become a carbon copy of your mentor either.
I agree that deliberate practice and acquiring tacit knowledge are not the same thing. To me, deliberate practice is about repeating a certain activity -- one that you typically can describe in words to someone who doesn't already know it -- enough times that it's available to you as a tool, eg playing scales as a musician, times tables in elementary school math. Tacit knowledge has more to do with how you decide to apply those skills to best effect.
But my experience has been that they have kind of a symbiotic relationship. If you didn't have some tacit knowledge to begin with, you wouldn't know what to practice, or when you had practiced enough to be good. At the same time, it may not be possible to acquire enough tacit knowledge to become an expert if you don't have an immediate command of certain skills developed through deliberate practice. I.e. there's feedback -- more tacit knowledge should make your deliberate practice more effective, and better skills make it easier for you to acquire tacit knowledge.
Actually, even the first two tables comparing the frequency of 1,2,3,4,5,6 when obtained using primes vs. a fair die suggest that consecutive primes do not give a truly random (uncorrelated) way of choosing congruence classes mod 7.
If I throw a fair die 10^6 times, the probability of getting any given single outcome should behave according to Poisson statistics. On average, if I repeat a trial of 10^6 die-throwings many times, the number of outcomes of "4" (let's say) should be on average 10^6/6 = 166,667 , as mentioned in the article.
However, the exact number of times "4" comes up in a given trial itself follows a distribution around that average whose spread is about sqrt(166,667), or about 400. So the typical "error" in the frequencies given in the table should be ~few hundred.
By this reasoning, the deviations in the top table, the one given by the primes, are surprisingly small -- of order tens rather than hundreds. In other words, primes are more equitably distributed among congruence classes than we would expect independent die roll outcomes to be.
He would sometimes eat lunch with us postdocs (just randomly, on a whim) when I was a postdoc at IAS. He would always have something interesting to say or ask, generally about science but often offbeat or unexpected -- you couldn't help but notice both his wide-ranging intellectual interests (our field was not quite within his main expertise) and his very gracious manner in starting conversations that both he and we would learn from.
There is a parallel with one of pg's other essays, 'Why nerds are unpopular'[1]. He says (I paraphrase): 'nerds care more about being smart than being popular'. But that's another way of saying nerds think the tests you have to pass to be popular are (on balance) a waste of time, ie bad tests.
As of earlier today you could still follow links from WSJ social media, for example their official twitter feeds, and get access to the whole article. Any way to set that up as a link in HN?
Rota was quite the celebrity, at least among students. In addition to advanced classes in his specialty, he taught differential equations, which was required for at least two thirds of undergraduates, so everyone knew him. Each semester he would pay a diffeq student to have a can of Coke ready for him at each class, so he could drink it while he lectured. Also, anyone who asked a question during lecture (a room of ~350 people) would get a free Hershey's bar after class. That he had worked in (and sometimes taught) philosophy as well as math was kind of the icing on the cake.
There was also a persistent rumor that he'd once given a diffeq multiple choice exam (scored like the SAT, so that a fraction of a point was deducted for every wrong answer) on which the average score was 1 or 2 percent, ie simply handing in a blank exam would probably have given a passing score. But I never confirmed that.
That's a pretty good 3-sentence summary - reflects my experience as well.
To expand a bit...it's not that nearly everyone doesn't start out super excited about their subjects and motivated by advancing human knowledge. It's that shortly post-PhD, you realize that in order to continue doing this work you find exciting and meaningful, you have to get money (for supplies, equipment, office space, not to mention your own salary), and the funding system is inherently kind of broken. It's not just the disappearance of tenure-track jobs. To caricature somewhat:
1. Funding is limited - grant success rates in my field are currently 10% to 15% - and in order to have a chance you need some history of having done interesting work, the more the better.
2. To carry out more of your ideas (to maximize the chances that some of them will turn out to be very interesting, as well as the overall amount and speed of work) you typically need more manpower than just yourself. You may also need equipment and people to operate it. Since you're probably a bit short of money, you hire trainees (students or postdocs), who cost less. Now you are really feeling the funding pressure since their livelihoods, not only your own, depend on your getting enough funds.
3. Eventually those trainees graduate, or become senior. Now they need to apply for their own grants and you need to find new trainees. In other words, the process of science today inherently increases the number of scientists competing for funding, and because each scientist during a single career typically requires many trainees, this number increases exponentially. There is no way the research funding budget can increase that fast over the long term.
4. The constantly increasing funding pressure means more and more people become preoccupied with their own survival above most any other professional concern. In addition to politics and ladder climbing, it hurts the science directly: if some project's not likely to get future funding, you might feel you don't have time for it, even if that's what interests you most about your area.
It's a hard problem...my impression is the reforms needed would be so sweeping that I'm not sure anyone has a complete picture of what things to do instead, let alone implement them.
Agreed, it can sometimes be exhausting. But I think being kind is (to be rather mercenary) very practical also: I've found it is much easier to get people to listen to and go along with opinions stated kindly (it is also exhausting to try to get through to people who've stopped being receptive because they're feeling attacked or shamed, or anticipating that feeling).
And it's definitely possible to disagree, even sharply, in a considerate manner - eg focus on the point of disagreement, not on the person disagreeing; avoid name-calling; point out that the issue you disagree with may be valid in other cases; etc.
What ivy league and other private research universities excel at academically is scholarly opportunities outside class assignments.
This is actually true at any school. At large state schools, there are professors doing real research. The difference with the ivy leagues is very few students actually seek these experiences out.
I completely agree (having been at a large state school) that there are many professors doing high quality research at state schools. I think the difference is (as you suggest) the funding and competition. Funding and even for-credit programs for undergraduate research were (I found) much more limited at state schools. You'd have 10 students, all with straight A's, applying for a single one-semester research assistantship that'd grant 1 course credit. At the same time I know of ivy school subject-specific fellowships of thousands of dollars for summer travel/study for which perhaps only 2 or 3 people would apply simply because there were so many other opportunities available. The net result is that if you're at an ivy, that kind of experience is much much easier to get.
I'm an academic - have worked at a variety of institutions but never officially been at an ivy league school, so I can't strictly speaking claim ground truth. But my impressions are, sure, if your idea of the "education" part of college is to (sometimes) go to classes required for your degree and do exactly what the instructor tells you and that's it, probably it doesn't make a huge difference in terms of academics where you went to school. Everybody uses (almost) the same textbooks, after all. And academic jobs are scarce enough that two instructors of the same subject at very different institutions may actually have had similar undergraduate educations and approaches to undergraduate teaching.
What ivy league and other private research universities excel at academically is scholarly opportunities outside class assignments. If you want to reconstruct 17th century pipe organs or build robotic insects or experiment on extremophiles in Greenland or whatever, there might very well be someone who wants a student to help with something like that. Plus they probably have money to cover your expenses and pay you a stipend for it, and you don't have to compete with 500 other people for the chance. More generally, if there is a particular academic topic you as a student want to learn all about, a top university has a better chance than most places of having someone or something that can help you. But the student has to show up with the initiative and persistence to get that.
The hard part is admissions: what's the best way to find students who will of their own volition seek this kind of thing out? The simple answer would seem to be to look for kids who have a history of doing that and succeeding. But instead that produced this arms race of people seeking lots of extra academic experiences purely for the sake of getting into college and then burning out and not wanting to continue once they got there.