Like many other students, I can’t deny that I spend sizeable amounts of my time on the Internet. I may not actively participate on any major forums, but I enjoy reading just about any such posts: it’s beneficial, because you can often pick up advice from people who are older than you and have more experience. I also feel there is much that can be learned from people younger than I am. Everyone has their own interesting perspective on things, and unique way of thinking.
A topic I’ve noticed a lot of interest in is how a person’s IQ relates to things like academic performance, the chances that person has at being successful in academia, and so on. Surprisingly, I’ve seen a lot of posts babbling things to the effect of “not everyone was created equal, only certain people are intelligent enough to do X”, and so on. It is of my personal opinion that this is, for the most part, nonsense. I’m not denying that there is some subset of living people who truly lack the capacity to, say, become mathematics professors. My point is that the last thing people should be telling these high-schoolers is “choose another field to work in, you probably wouldn’t make it in that.” Is this really the kind of thing we want to be pouring into ambitious children’s ears? Sure, in today’s digital society, it seems like everyone craves numbers and statistics. Scoring a high number on an IQ test doesn’t make you better than anyone else, nor is it a reliable estimator of how much you’ll achieve in life, or how fat your salary will be. There are plenty of people who are freak geniuses on paper and still contribute little to nothing notable to society. Let’s face it, the people who prance around displaying their membership to some “High IQ Society” are insecure, arrogant, and in general, you probably don’t want anything to do with them. Hard work will get you far, and that’s more or less the recurring theme of this post. To anyone reading this who is thinking of entering mathematics, I offer all my encouragement. Don’t think that you’re unfit just because you didn’t spend your childhood training for math contests.
In late high school, after correspondence with fellow students and a few books, I realized what mathematics was. It wasn’t about calculating dot products of vectors, or finding solutions to quadratic equations, or any of the other computational nonsense that pollutes the present-day Canadian high school math curriculum. It was about logic, deduction, structure, and beauty. From this point on, after reading from a couple of books that didn’t have too many pictures in them, I became cynical of everything. I didn’t particularly care about anything my Calculus teacher said; I learned the methods, the equations, whatever else I needed to get the marks in the course. And on the side, through these books, I was giving myself a small supplement of the superior mathematical background endowed upon many who came before me. Canada’s school system has folded. There is no longer anyone concerned about teaching critical thinking, abstract reasoning, or other important skills. It’s as though the country is begging to be left in the dust of intellectual progress. As students here are encouraged to be academically lazy, the theorists of tomorrow trickle out of schools at an ever-slowing rate. All the while, elsewhere in the world, the exact reciprocal is taking place.
This brings me to my next topic: success in university. It is not natural that someone can attend class all term, occupy themselves only marginally with their course material, and score near-perfect on exams. I’m not one of the “effortlessly near-perfect” people I mentioned. I’m one of many students who is burdened by making careless mistakes on exams or being presented with the odd exam problem that seems to be “just beyond” my reach. A lot of people are in search of some academic success secret. After searching for it myself, I feel ready to announce it. The good news is that the secret is very straightforward. The bad news is that for most people, this lifestyle change is not one that takes place easily. After enough analysis of my own learning patterns, I’ve concluded that the solution is solitary exploration. Studying with others is great in certain situations, but what I’m talking about is something you have to do alone, and most likely in front of a blackboard after class hours are over or something (unless you really don’t mind using up stacks upon stacks of paper). In the average day, a university student learns plenty to keep him or her occupied for a few hours just exploring the concepts, testing the methods learned, and so on. It’s almost spiritual. What class of problems can be solved by a given method? Can you show beyond a reasonable doubt that any problem solvable by a given method must be of that class? If not, perhaps you are missing something, and the method is more broad than the examples suggest. If this is the case, construct new problems; problems that have fundamental structural differences from the ones given to you. Try to solve them. If you have time, develop your own way of solving them, just for fun. If you’re in math like I am, come up with theorems about stuff – anything that seems intuitively plausible to you – and try hard to prove it. If you can’t, maybe you should try looking for a counterexample. You literally have to get your hands dirty, feel every ripple, every bump, every scratch in the concepts you’re studying. Become uncomfortably intimate with them. My point is that practicing things like this will train you a lot better to think critically about the material, the type of thought that often eludes you until the exam rolls around. Poking around in the dark forces you to develop some pretty hardcore concept-synthesizing skills. Assignments help with this, but at least in my case, my own exploration pays off a lot more. If you’re not being challenged by the assignments, this isn’t good either, and you should find some alternative source of problems that will truly push you beyond your comfort level.
Remember: if you’re not spending a significant amount of time each week struggling through problems, then you’re doing the wrong problems. The pain is necessary, and there is no alternative. So eat dinner, go to campus, find an empty lecture hall, and cover some blackboards. Godspeed.