I remember when I first took a class in linear algebra, and we were talking about vector spaces. In addition to the definition of a vector space, we were also given multiple axioms that define what the structure of a vector space looks like. This included a bunch of boring things, like the fact that if you have a vector v, you should have a corresponding vector -v such that v+(-v)=0, where 0 is the zero vector. There are eight of these axioms, and together they describe exactly what can be called a vector space.

There’s a saying among students regarding preparing for an exam. In short, it goes like this: Study a lot before the test, and then you can forget most of what you know.

Everyone solves problems differently. Some like to work directly with the mathematics head on, while others prefer to have a more intuitive approach. This includes studying simpler cases of a problem, or looking at examples in order to really understand what’s happening. These are all valid approaches, but the point I want to highlight is that these are all strategies. There’s a certain method to tackling a problem. It’s not that you can’t solve a problem through trial and error, but if you want to solve more problems more quickly, your best bet is to figure out a strategy.

Why is it that some students will pick up new subjects in mathematics and science relatively easily, while others will struggle for weeks on end getting just the simple concepts down? As a tutor, I see this all the time. In fact, I would almost wager to say that most students I work with aren’t actually having difficulty with the topic they say they don’t understand. So what’s going on?