By Tai-Danae Bradley
A while ago at my blog Math3ma, I wrote a post in response to a great Slate article reminding us that math – like writing – isn’t something that anyone is good at without (at least a little!) effort. As the article’s author put it, “no one is born knowing the axiom of completeness.” Since then, I’ve come across a few other snippets of mathematical candor that I found both helpful and encouraging. And since final/qualifying exam season is right around the corner, I thought it’d be great to share them here for a little morale-boosting.
The first comes from a fantastic post written by University of Illinois at Chicago’s recent PhD Jeremy Kun (also blogger at the excellent Math ∩ Programming) in which he answers the question What is it ‘really’ like for a mathematician to learn math? In short, his answer is contained in the post’s title: “Mathematicians are chronically lost and confused (and that’s how it’s supposed to be).”
Of course, he means this in a good way and elaborates by sharing this colorful metaphor of mathematical research by Andrew Wiles:
You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are a culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them.
I definitely feel like I’m stumbling in the dark whenever I tackle a new subject. (I believe the technical term for this is learning.) And I’m not even at the research stage yet! In response to this occasional feeling of lostness that we students tend to feel, Kun says:
Sometimes the answer is to pinpoint one very basic question I don’t understand and try to tackle that first. Other times it’s to learn what I can and move on.
This is great advice for graduate students. Personally, I find that there’s just not enough time to learn it all, so often I have to be okay with not being able to understand a topic as well as I’d like. But occasionally I end up “wasting” an entire day just grappling with a single problem/idea. Although those days are frustratingly slow, they actually end up being the most productive and satisfying and not a waste at all! So I was encouraged when Kun mentioned that one of his colleagues has similar sentiments:
If I spend an entire day and all I do is understand this one feature of this one object that I didn’t understand before, then that’s a great day.
A great day, indeed. In fact, I find that math often comes with an initial “shock factor” that (usually!) goes away once I make a conscious decision to focus and, well, do the work. It’s sort of like going for an early-summer swim at the ocean or pool. The water looks inviting, but as soon as you jump in, its unexpectedly frigid temperature takes your breath away, and your only thought is Agh! What am I doing here?! But after wading around a bit, your body adjusts and the cold doesn’t feel so bad anymore. And that’s when the real fun begins.
Case in point: I was recently working through some of my school’s old topology qualifying exams and came across a question which I initially had no idea how to solve. In fact, here’s the problem (You don’t actually have to read it!):
But here’s what my brain saw:
Yep. You see, my first thought was, Is this even English?! But after a few minutes of pondering, I realized just how simple both the question and its solution were! Of course, in other instances, minutes turn into hours, days, or even semesters, and the end result doesn’t always turn out to be simple. But in general, I find that once I put in the work, the math isn’t as terrifying as it seems at first. And as Wiles observed, it’s okay – necessary, even! – to stumble around in the dark before finding the light switch.
Pretty encouraging, right? Well it doesn’t end there! I’ve got more quotes from great mathematicians that I’d love to share with you. So be sure to stay tuned for part 2! In the meantime, I’ll leave you with a few links that touch on some of the things we’ve talked about today:
- This Quora answer to “What is it like to understand advanced mathematics?” and this article from the Princeton Companion to Mathematics emphasize the amount of hard work one must do in order to produce good results.
- Here’s a pretty funny but all-too-real answer to the question “What do grad students in math do all day?”
- At the 2013 Joint Mathematics Meetings, Francis Su gave an excellent talk on why teachers should not value students based on their accomplishments and academic performance. I also love this article he recently wrote for the MAA Focus news magazine.
- Speaking of performance, both Cathy O’Neil and Dan Lee have written about the ugly side of math competitions.
- And though not math-specific, this tweet needs to be tattooed to the backs of eyelids of students everywhere. (Or at least framed and hung on a wall!)
This article was adapted from Bradley’s Snippets of Mathematical Candor, originally published at Math3ma on April 4, 2016.