aardvarks

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.

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.