A Forgotten Equation
Yesterday, I overheard some members in a research group where I hang out, expressing a fleeting uncertainty over how a paper performed a Poincare compactification of ℝ (well, ℝ^n).
In terms of our use case, it was about how to build an order-preserving self-map on the real line such that -∞ goes to -1, 0 stays at 0, and +∞ goes to +1. Compared to all the real dynamics that went on in the discussion, this was a trivial task. Nonetheless, eager for (any) chance of doing (any) maths with (any) other people, I jumped in.
I recalled that there would be a unit circle somewhere in the geometric representation of the map, but I couldn’t pinpoint where to put it until the way after the end of the meeting, by which point I was late for a class I am auditing.
“You shouldn’t worry too much about it, and you’d better go to class now,” one of the professors told me, “Feel free to work this map out on your own if you like it, just keep in mind it’s been common knowledge in maths for a long time.”
Of course I knew that this is common knowledge in mathematics too. Still, for some reason, her words tugged a string.
At first it was a sense of regret to have forgotten a fundamental equation that I’ve learned before, as I more and more often have.
Then it changed to suspicion, an insecure internal query if this kind of research life is the one that I really want to lead — if renowned mathematicians ought to collectively forget about a simple formula.
At long last, my overreaction to this little “nothing new” problem might have paid off, as I write here, trying to document a profound change in my outlook of learning and research.
Mental Arithmetic and Self-Image
Up to the recent conclusion of my undergraduate career — I have not really collected any meaningful data after — from high school A-Level tests, to university final exams; from the regional chemistry competition, to the dreaded Physics GRE, I have not referred to the provided sheets of formulae during an exam, likely ever.
The sense of achievement from beating such a self-imposed “challenge mode”, alongside a feeling of superiority when my Shandong-trained manual trigonometric expansion and mental calculus, etc, beat my calculator-wielding Western classmates, became the main factors to which I subconsciously accredit my pleasant student life.
Gradually, as I now realize, much of my confidence in life was built on a sense of fluency in my fields of study and academic self-reliance. My life is, like all other STEM students’, full of after-class discussions, web searches, number crunching, black-boxing, and StackExchange questions.
Though many friends tried talking me out of the this — thank you — hardly any of these mattered to me more than exams and GPA for a long time. Surely, sans a few malfunctions, such a mentality was one that worked for me — I wanted a good-looking transcript, and got one.
As alluded prior, there were two times — among others — that I could’ve looked to the other way and probably learned more from a course / gotten better grades in an exam. I thought my brain had simply malfunctioned then, but now I believe they were indeed symptoms of something deeper, the other side of this story about a life in physics.
Interestingly, these two occasions were back-to-back physics exams at UC Berkeley, and the only two exams that I’ve saddened myself with by that point —
have you kicked the door to 50 Birge so hard it got damaged? Before, I struggled to dismiss them as some hopefully inconsequential six hours in my career. But now, I am glad that I did remember them.
The first was my Physics 112, Statistical Mechanics. It is the only exam I’ve taken where a self-prepared cheat sheet was required. I didn’t quite know what to put on it, wrote something perfunctory in embarrassingly large fonts, and — as you might have guessed — put it away during the exam.
As it soon turned out, I didn’t remember as much from the first half of the course than I thought I did, and missed out on some key “first-steps” to the final problem, and had to throw the entire solution away.
I have written those definitions and equations all down in my course notes, my post-mortem found, but I was too arrogant to spare a second look at them before the exam.
The second —
it took place immediately after 112 across the campus, so please excuse me for kicking the door — was Physics 139, General Relativity. Blimey, that was the only open-book exam in my student career so far. I remember that some questions were quite open ended, or contained descriptions of space time models / wormholes that I was supposed to construct a model of. That was not the kind of thing I was preparing for. Probably, if I’d spent the time indexing the textbook, familiarizing myself with the actual way to abstract a spacetime model into a PDE set, all the equations I’d remembered may be useful then.
I should have treated this observation with more of the scientific rigor that I pride myself in: remembering all equations do not guarantee understanding of the content, nor ability to solve problems. If one remotely makes the memorization of equations a standard or even a goal, then one might as well start missing out on the exciting stuff in research.
This is not big news, but saying it out loud helps: my career ahead will not be a closed-book exam. And, whether my closed-book exam track records do prove that I’ve learned what I should from those exams
— some people who’s seen my transcripts clearly said no — I need to be extremely careful to decide for myself.
A Nonessential Middle Ground
I’ve oversimplified quite a lot. In fairness, I’ve been definitely smart enough to solve all the book exercises and most of the exam problems thrown at me in my life. That said, chances are, in my speedy production of solutions, I sometimes neglected the real learning process, and, worse, the actual intended learning outcomes, to the extent that my research life needs deep introspection to kickstart.
We laugh at the notion that “professional mathematicians are people who multiply big numbers together”, but maybe not as much at “professional mathematicians memorize complicated equations.” However, as we’ve seen now, neither is a good standard to measure intellectual sophistication.
Some “complex-looking” equations just show up in one’s life and stays there after ample exposure and practice, like the Laplacian operator in 3D spherical coordinates — it looks horrendous, but I know it by heart —, or the Maxwell equations.
Some others represent deep relations to grasp — there’s plenty, don’t get me wrong — and a mathematician does handle complex objects as his/her job. But more to the story is a cognizance of priorities, and an open mind that recognizes what is known — possible to be looked up — and what requires collected efforts to become known.
When I was doing my Year 10 Olympiad physics in China, I was struck by both the number of disjoint skills and relations one needs for the exam, and the amount of exercise problems one needs to clear every week. My horrendous time management then shifted my attention to the former.
It wasn’t, arguably and evidently, the best strategy in the business.
Thank goodness I did not fall far behind. Famous last words?