Over on Google+, a computer scientist at McGill named Artem Kaznatcheev passed on this great description of what it’s like to learn math, written by someone who calls himself ‘man after midnight’:
The way it was described to me when I was in high school was in terms of ‘levels’.
Sometimes, in your mathematics career, you find that your slow progress, and careful accumulation of tools and ideas, has suddenly allowed you to do a bunch of new things that you couldn’t possibly do before. Even though you were learning things that were useless by themselves, when they’ve all become second nature, a whole new world of possibility appears. You have “leveled up”, if you will. Something clicks, but now there are new challenges, and now, things you were barely able to think about before suddenly become critically important.
It’s usually obvious when you’re talking to somebody a level above you, because they see lots of things instantly when those things take considerable work for you to figure out. These are good people to learn from, because they remember what it’s like to struggle in the place where you’re struggling, but the things they do still make sense from your perspective (you just couldn’t do them yourself).
Talking to somebody two or levels above you is a different story. They’re barely speaking the same language, and it’s almost impossible to imagine that you could ever know what they know. You can still learn from them, if you don’t get discouraged, but the things they want to teach you seem really philosophical, and you don’t think they’ll help you—but for some reason, they do.
Somebody three levels above is actually speaking a different language. They probably seem less impressive to you than the person two levels above, because most of what they’re thinking about is completely invisible to you. From where you are, it is not possible to imagine what they think about, or why. You might think you can, but this is only because they know how to tell entertaining stories. Any one of these stories probably contains enough wisdom to get you halfway to your next level if you put in enough time thinking about it.
What follows is my rough opinion on how this looks in a typical path towards a Ph.D. in math. Obviously this is rather subjective, and makes math look too linear, but I think it’s a useful thought experiment.
Consider the change that a person undergoes in first mastering elementary algebra. Let’s say that that’s one level. This student is now comfortable with algebraic manipulation and the idea of variables.
The next level may come somewhere during a first calculus course. The student now understands the concept of the infinitely small, of slope at a point, and can reason about areas, physical motion, and optimization.
Many stop here, believing that they have finally learned math. Those who do not stop, might proceed through multivariable calculus and perhaps a basic linear algebra course with the tools they currently possess. Their next level comes when they find themselves suffering through an abstract algebra course, and have to once again reshape their whole thought process just to squeak by with a C.
Once this student masters all of that, the rest of the undergraduate curriculum at their university might be a breeze. But not so with graduate school. They gain a level their first year. They gain another their third year. And they are horrified to discover that they are expected to gain a third level before they graduate. This level is the hardest of them all, because it is the first one that consists in mastering material that has been created largely by the student.
I don’t know how many levels there are after that. At least three.
So, the bad news is, you never do see the whole picture (though you see the old picture shrink down to a tiny point), and you can’t really explain what you do see. But the good news is that the world of mathematics is so rich and exciting and wonderful that even your wildest dreams about it cannot possibly compare. It is not like seeing the Matrix—it is like seeing the Matrix within the Matrix within the Matrix within the Matrix within the Matrix.
As he points out, this talk of ‘levels’ is too linear. You can be much better at algebraic geometry than your friend, but way behind them in probability theory. Or even within a field like algebraic geometry, you might be able to understand sheaf cohomology better than your friend, yet still way behind in some classical topic like elliptic curves.
To have worthwhile conversations with someone who is not evenly matched with you in some subject, it’s often good for one of you to play ‘student’ while the other plays ‘teacher’. Playing teacher is an ego boost, and it helps organize your thoughts – but playing student is a great way to amass knowledge and practice humility… and a good student can help the teacher think about things in new ways.
Taking turns between who is teacher and who is student helps keep things from becoming unbalanced. And it’s especially fun when some subject can only be understood with the combined knowledge of both players.
I have a feeling good mathematicians spend a lot of time playing these games—we often hear of famous teams like Atiyah, Bott and Singer, or even bigger ones like the French collective called ‘Bourbaki’. For about a decade, I played teacher/student games with James Dolan, and it was really productive. I should probably find a new partner to learn the new kinds of math I’m working on now. Trying to learn things by yourself is a huge disadvantage if you want to quickly rise to higher ‘levels’.
If we took things a bit more seriously and talked about them more, maybe a lot of us could get better at things faster.
Indeed, after I passed on these remarks, T.A. Abinandanan, a professor of materials science in Bangalore, pointed out this study on excellence in swimming:
• Daniel Chambliss, The mundanity of excellence.
Chambliss emphasizes that in swimming there really are discrete levels of excellence, because there are different kinds of swimming competitions, each with their own different ethos. Here are some of his other main points:
1) Excellence comes from qualitative changes in behavior, not just quantitative ones. More time practicing is not good enough. Nor is simply moving your arms faster! A low-level breaststroke swimmer does very different things than a top-ranked one. The low-level swimmer tends to pull her arms far back beneath her, kick the legs out very wide without bringing them together at the finish, lift herself high out of the water on the turn, and fail to go underwater for a long ways after the turn. The top-ranked one sculls her arms out to the side and sweeps back in, kicks narrowly with the feet finishing together, stays low on the turns, and goes underwater for a long distance after the turn. They’re completely different!
2) The different levels of excellence in swimming are like different worlds, with different rules. People can move up or down within a level by putting in more or less effort, but going up a level requires something very different—see point 1).
3) Excellence is not the product of socially deviant personalities. The best swimmers aren’t “oddballs,” nor are they loners—kids who have given up “the normal teenage life”.
4) Excellence does not come from some mystical inner quality of the athlete. Rather, it comes from learning how to do lots of things right.
5) The best swimmers are more disciplined. They’re more likely to be strict with their training, come to workouts on time, watch what they eat, sleep regular hours, do proper warmups before a meet, and the like.
6) Features of the sport that low-level swimmers find unpleasant, excellent swimmers enjoy. What others see as boring – swimming back and forth over a black line for two hours, say – the best swimmers find peaceful, even meditative, or challenging, or therapeutic. They enjoy hard practices, look forward to difficult competitions, and try to set difficult goals.
7) The best swimmers don’t spend a lot of time dreaming about big goals like winning the Olympics. They concentrate on “small wins”: clearly defined minor achievements that can be rather easily done, but produce real effects.
8) The best swimmers don’t “choke”. Faced with what seems to be a tremendous challenge or a strikingly unusual event such as the Olympic Games, they take it as a normal, manageable situation. One way they do this is by sticking to the same routines. Chambliss calls this the “mundanity of excellence”.
I’ve just paraphrased chunks of the paper. The whole thing is worth reading! I can’t help wondering how much these lessons apply to other areas. He gives an example that could easily apply to mathematics—a
more personal example of failing to maintain a sense of mundanity, from the world of academia: the inability to finish the doctoral thesis, the hopeless struggle for the magnum opus. Upon my arrival to graduate school some 12 years ago, I was introduced to an advanced student we will call Michael. Michael was very bright, very well thought of by his professors, and very hard working, claiming (apparently truthfully) to log a minimum of twelve hours a day at his studies. Senior scholars sought out his comments on their manuscripts, and their acknowledgements always mentioned him by name. All the signs pointed to a successful career. Yet seven years later, when I left the university, Michael was still there-still working 12 hours a day, only a bit less well thought of. At last report, there he remains, toiling away: “finishing up,” in the common expression.
In our terms, Michael could not maintain his sense of mundanity. He never accepted that a dissertation is a mundane piece of work, nothing more than some words which one person writes and a few other people read. He hasn’t learned that the real exams, the true tests (such as the dissertation requirement) in graduate school are really designed to discover whether at some point one is willing just to turn the damn thing in.
Such a wonderfully wise post – I wish someone had told me all this thirty years ago! Thanks for another fascinating insight John.
Thanks! I wish someone had told me this thirty years ago, too! If I had a time machine I could teach my younger self so much.
I think one’s humility increases with age.
Eh. This only takes you so far. These excellent swimmers have different technique because their abnormal bodies and metabolism allow them to do so. They find the horrible routine exciting because they succeed (relatively) effortlessly at it. If you grab an average person off the street and try to make these techniques succeed for them, you will be quickly disillusioned.
No matter what the level, each of us has probably repeatedly experienced the profundity of the maxim:
“If I have seen a little further it is by standing on the shoulders of Giants.”
Till recently, I was unaware that prior to Isaac Newton’s (above) tribute to Rene Descartes and Robert Hooke in a letter to the latter, it was reportedly the 12th century theologian and author John of Salisbury who recorded in Latin an even earlier version of this humbling admission—the gist of which is translatable as:
“Bernard of Chartres used to say that we are like dwarfs on the shoulders of giants, so that we can see more than they, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant size.
Perhaps, what Bernard of Chartres intended was to suggest that it doesn’t necessarily take a genius to see farther; only someone both humble and willing to:
(i) first, clamber onto the shoulders of a giant and have the self-belief to see things at first-hand as they appear from a higher perspective (achieved more by the nature of height—and the curvature of our immediate space as implicit in such an analogy—than by the nature of genius); and,
(ii) second, avoid trying to see things first through the eyes of the giant upon whose shoulders one stands (for the giant might indeed be a vision-blinding genius)!
If so, today I can more fully appreciate that, in my case, it was the latter lesson that I was incidentally taught by—and one of the few that I learnt (probably far too well for better or worse) from—one of my Giants, the late Professor Manohar S. Huzurbazaar, in my final year of graduation in 1964.
This is a trillion dollar question. Are people stupid or do they act stupid because it is more comfortable, I could never figure out that one. Not even with my own children!
The strange thing is that these people who act so stupid in their “career” seem to be almost genius in every other stupid activity, really makes you wonder.
Interesting. I arrived at the three-level phenomenon from the perspective of performance professions like music and sports. But it seems reasonable to expand to include many academic fields as well. One can divide, say, music or sports into three levels. First, the amateurs – they do it for fun. We onlookers watch them and say, “They are not bad, with some hard work maybe I could do that.” Then we look at the pros and say, “They are good! They are the Stars of the field! I could never do that!” Finally, every 10 or so years, a Super Star emerges. We look on and marvel at an effortless performance; “That looks easy! That can’t be too difficult!” But the Stars look on is disbelief, “That’s impossible. If only I could do that!” Brings to mind a Feynman taking an unknown student’s thesis, riffling through it back to front in a few seconds, and then suggesting several improvements along with a general summary. That looked easy, it can’t be too difficult…
There is an anecdote in ‘Surely You’re Joking, Mr. Feynman’ wherein Feynman was describing to a visiting Fermi a complex situation in which Feynman’s ability to qualitatively grasp the essence had been stymied, and he’d had to wait for the calcs.
Right after the setup Fermi said “wait, don’t tell me – it’s going to come out like *this* [he was right], and the *reason* it’s going to come out like this is blah blah blah.” Feynman said that encountering someone who could do what he was usually best at, but 10 times better, made a big impression on him.
Hmmm… Maybe a generalization is in order. Define n-Stars. n = 0 is a Star, n = 1 is a Super-Star, n = 2 happens a couple of times a century, and I can’t even imagine what n = 3 is. Kind of like n-Categories. :-)
n=3 is Wolfgang Amadeus Mozart
Another thing I’ve noticed about these sort of “levels” is that they come with increasing error tolerance. For a student learning basic algebra, reversing a sign might be a serious enough error to cause them to get stuck on the problem and have a lot of trouble figuring out where they went wrong. At a higher level, it’s often glossed over with a remark like, “Oops, that should be negative — okay, do the same steps and it works out similarly,” or even, “up to flipping some signs around somewhere” if it really doesn’t matter.
Higher-level reasoning is less sensitive to minor differences — partly because of a better ability to see which differences are truly minor. I think that’s the main reason why small errors can become less important.
Yes, seeing that certain classes of errors are very likely to be easy to fix means you can charge ahead and get good results much faster, then go back and fix up the details. This is incredibly important in math; I’m not so familiar with this phenomenon in other subjects, but it’s probably rather general. (Politics? I can imagine a savvy politician saying “don’t worry, we can deal with that later.”)
I would guess it would apply quite generally in areas where one can always go back and fix a mistake. But in some areas (like chess or Scrabble) where decisions are irreversible, even “small” or lower-level errors could be quite punishing.
I think that in chess as played by humans, higher level thinking is
-critically *necessary* (e.g. to select appropriate high level goals, to quickly find good candidate moves based on position type, and to radically prune the move tree by immediately recognizing certain positions as won or drawn w/o further analysis)
-but not *sufficient* (because you can do all that stuff real well, but if while analyzing move 14 you miss that tactical gotcha in a critical variation at move 19, it’s very likely that your grandmaster opponent will not:-).
I’m currently doing my PhD in theoretical physics and this is something that I feel that I have just started learning. Before it was impossible for me to finish lengthy calculations without any reference material or help because I would get lost in finding and fixing small errors (like signs being wrong). Nowadays I’m much better at seeing which parts of the calculation are truly important and which are just details. As a result, I can (for a moment) just ignore the errors I make in the details and focus on the important stuff to get a result that is at least almost correct. Furthermore, I can often detect and correct the errors in the important parts of the calculation by heuristic arguments.
I hope I will get even better at this. It seems to me that the experienced physicists often just deduce or guess their results from some heuristic arguments and then do the math just to make sure they got it right. Currently this seems like black magic to me.
Someday you will be a magician too. The key is to constantly develop your list of heuristics: be very conscious of what they are, learn new ones, test them, find the exceptions, try to generalize them, think about what happens when two come into conflict, think about the list of meta-rules I’m stating now and keep refining that, and so on. Eventually you’ll be able to follow John Wheeler’s rule: never do a calculation unless you already know the answer.
Thank you for the acknowledgement, John! I am glad you are getting this view of mathematics out to a wider audience. I really like the comparison with more physical pursuits, and the reminder that it is not just one ladder, but a ladder for each concentration or approach. However, I do think that there are a few steps to ‘mathematical maturity’ before the countless other ladders of mathematics become accessible. Unfortunately, it seems that most people are turned off math before they get a chance to take even these first few steps :(.
This notion, or a similar one, has been systematized in Human Resources Management: https://en.wikipedia.org/wiki/Competence_%28human_resources%29
It is actually quite useful to express in writing the progression from beginner to expert in a given discipline. A similar concept, named “maturity model”, can be used to, for instance, evaluate how far along an organization is to mastering the implementation of complex software management projects.
I am also reminded of this meme in relation to people who stagnate at the same level: “Some people don’t have 30 years experience. Merely 1 year, 30 times over”.
I want to know more about the cover photo for this article. Those vertical stone stairs look like they have an interesting story. Where are they?
I found this image floating around the internet. The name of this file is
So, these are some steps going up Hua Shan, outside Xi’an. ‘Shan’ means ‘mountain’. And if you Google that, you’ll get to Wikitravel’s article on Mount Hua. It’s quite a place, one of China’s traditional Five Great Mountains:
It has very steep steps for pilgrimages:
Climbing at night because it’s safer to simply be unable to see the extreme danger!
Brings back fond memories. I had the good fortune to be there in 2004. Both the view and the climb were breathtaking! The lines of people going up and coming back were endless, from sunrise to sunset. Amazing views, countless people. China. Thanks for the pictures and the history lesson, John.
This is very interesting. Let me ask a slightly related question: How do you get back to the level you were once on?
I ask because I am finding the theory of differential equations very difficult right now, and it seems to be because I’ve lost so much of what I learned about analysis nearly a decade ago. Being humbled on occasion is, I suppose, a good thing, but I’ve had enough of it!
I’ve never quit working on math for a long period of time, so I’m probably not the right one to answer this question.
I have quit working on particular branches of mathematics, sometimes for years, and then picked them up again. Here I’m helped somewhat by the fact that I write extensive explanations of many things I’ve learned. So, when I go back to a subject, I reread what I’d written about it! You probably can’t do this. The other thing I do is just think about a topic very hard for a few weeks, and do lots of calculations and other scribbling. At first it seems hard, but usually it gets easier with time. And often I find that forgetting things helps me move to a new higher level of understanding, because unimportant details wash away while the big picture remains, and I can fill in that big picture in a new, better way.
Anyway, it’s an interesting question, and I hope some other people answer it—especially people who have quit math altogether for years, and then resumed it!
After spending an hour reading a high school alum’s blog and this quite motivational blog post, I have determined that the reason for my total and utter failure during midterms, at least from a math perspective, is basically not enough math…
Speaking of stair analogies, I like to regard think of mathematical problems as a spiral staircase. Very often, it looks like I have returned to where I started, but the important thing is to figure out whether I have come back at a higher or a lower level.
[…] “I don’t know how many levels there are after that.” […]
Loved this article, initially discovered, where else, on marginalrevolution.com, and especially this part (emphasis added):
There was a spike of hits on this blog article today due to a link from marginalrevolution.com. It peaked at slightly over 1000 hits per hour, about ten times the normal rate:
It could be good to find out how to get as many people to read Azimuth as read that website. Or maybe not.
Hmm, and yesterday’s spike was dwarfed by today’s:
So far today 24,048 people looked at this post, as compared with 3,438 yesterday. It’s still marginalrevolution.com that’s causing it. I’m curious about these spikes in attention, what causes them and what their effects are. It seems clear that a number of people (including me) enjoyed this passage by Artem Kaznatcheev on levels of excellence in math:
Reblogged this on Pink Iguana.
A remarkable set of nested blog posts. I’ll quote one piece but it really is worth reading it all […]
Computer programming is EXACTLY like this.
In my opinion, you are being reductionist to an extreme.
If someone is still there and has not finished their PhD there is a world of posible reasons. You could just dig a little deeper to know or you can just invent a plausible generalizarion based on your incomplete knowledge of the situation.
You choosed the second one.
Just curious – what is James Dolan doing these days?
I believe he’s in Long Island living at his mom’s place, thinking about algebraic geometry and other things.
Is his mom an algebraic geometer?
I just stumbled upon this great blog post on “levels of excellence”. The author uses material on mathematicians and professional swimmers, but there are many […]
[…] Levels of excellence. Interesting view on learning. It is fascinating how sometimes it truly seem like “a level”, a real qualitative change. […]
Reblogged this on computerCalledVarun().
thanks, this was a great read. i like to summarize this way:
“excellence is a side effect of hard work and discipline. success is a side effect of excellence.”
[…] but a friend linked me to a new mathematics blog today and my experience was very similar to this post in which Baez talks about “levels of excellence. I love his idea of playing “teacher/student”. I’ve done that so many times […]
[…] Levels of Excellence | Azimuth: From the same blog. Interesting pointer to a paper by Daniel Chambliss (“The mundanity of excellence”). […]