This is entirely based on a post from one of my favorite blogs: Math With Bad Drawings. Read the original post if you wanna see really cool drawings. The post starts with a simple question (and the author provides a lovely answer):
Does each person have a math ceiling? (or a cognitive ceiling of any kind)
Answers usually fall in one of 2 categories. The optimist claims that there is no such thing and everything is achievable by enough work and determination. Some might get it faster than others, but there is no limit. The pessimist claims the opposite: No matter how much one works there are things beyond some people’s comprehension (I’ve heard this mostly from teachers and tutors who taught hundreds of students, with astonishingly disparate performances). Some middle ground can be achieved (someone claiming that there’s no math ceiling as long as you find the right circumstances, including the right teacher, the right mood and motivation to work,…). Still, these are the 2 main campuses of thought.
And then comes Math with Bad Drawings. They start with a beautiful analogy: Say I’ve bought a futon for my room. As I was getting it in, I saw a small metal piece come off. I’m not sure where from, and after looking around and everything seeming to be in place, I have no choice but to take it into my room as planed. All is fine. A week in and the futon starts bending down in the middle. As more time progresses the bending gets worse and worse, to the point of me not being able to sit in it anymore without sliding to the center (and everyone else sitting down too, into a awkward cluster of people).
And so it is with maths.
When we are learning maths we are supposed to follow the rhythm of the class. When we fail to do so we take shortcuts in order to do as everyone else, prioritizing implementation over understanding. We might be able to differentiate equations to find maxima/minima, but that’s useless unless we understand that the derivative is nothing but the slope of the curve at that point and as such the zeros of the derivative give us the maxima/minima. This is obvious to the half of the class that never took shortcuts in previous years and tried to understood everything. For the other half, they’ll feel like they’re just not capable of understanding the concepts involved, and going back to relearn the misunderstood concepts properly is a lot more work than learning them correctly the first time around. As Math With Bad Drawings said, a student who doesn’t understand a topic is a student with an expiration date on what he’ll be able to do.
My final remark to this is the following: it is easier to be an A* student for the entirety of your studies than to be a B or C student for the same period. So put on the effort and climb up the ladder: never take a “You’ll understand it once you do it enough times” for an answer, first understand it and them practice implementing it.
PS: I kind of feel really bad for taking the punchline “And so it is with maths” word by word from Math With Bad Drawings, but it is a brilliant punchline, I couldn’t deprive people of that. Sorry Math With Bad Drawings.