Why Malaysians Have No New Ideas
One annoying thing about studying at an elite university in the United States is that they don't necessarily recognise your A-Levels or STPM credits — meaning you have to take courses covering material you already mostly know. Somewhat disappointingly, I've found myself in this position when it comes to mathematics — it's handy to have a refresher of material you learnt two years ago, but agonising when most of your classmates need extra examples in class because it's completely new.
It's of course not all bad. A handful of material in the course is completely new to me, which is why I decided to avoid trying to place out of it. I might be bored 90% of the time, but knowing the material like the back of my hand can only help my grade, and the other 10% of the time, I might actually learn something.
Another advantage, naturally, is that I get to compare the teaching methods used by the American education system with those we use in Malaysia. My experience with the calculus we learn in this course stems mainly from my experience with A-Levels, because I skipped form four and form five (a bit of a long story, that). Considering that in many areas, the A-Levels go beyond what this course covers, I was not expecting to learn much in these fields.
What surprised me is that a number of things I struggled with in my A-Levels course have been remedied here. At this stage of learning in mathematics, the main skill is memorising basic formulae and learning how to plug the right things into these formulae.
The Malaysian method does not really help much when it comes to this. You learn things like the product rule, quotient rule and chain rule, but it does not really seem clear where and when you apply them. You use them as a matter of blindly following the process you're supposed to follow.
Initially the American course seemed largely the same. What impressed me, however, is how lucidly the instructor made it clear when she was applying the rules we had been taught. In Malaysia, the teachers would tell you what they were doing, but only in the most obvious sense. Sure, we can see you're differentiating what's inside the parentheses, but we don't know why you're doing that — we just know that's what you're supposed to do. It's rather agonising having to individually memorise the fact that you're supposed to differentiate the arguments to the natural log and trigonometric functions, especially when you consider that all you really have to know is that in all these cases the chain rule applies, and how to use the chain rule.
The other thing that surprised me was how the concept of calculus was explained and justified to us. In Malaysia, we went right away to the power rule, without any explanation of what a derivative is beyond the barest minimum, i.e. that it is the gradient of the function being differentiated. In the US, I was treated to a lengthy explanation of the concept of limits and how they led to the development of calculus. Knowing what a derivative fundamentally is, instead of blindly memorising a definition in vague terms you do not understand, proves surprisingly helpful in studying calculus (and makes it slightly less of a chore).
Even before I learned this approach to mathematics, I already had my doubts about mathematical education in Malaysia. As I've said before, it is one of the less atrociously-taught subjects in our schools, yet I have often gotten the sense that instead of understanding what we are taught, we only gain a vague definition of a word we do not understand in more words which we do not understand.
I experienced this particularly profoundly in statistics. Most of my classmates seemed to approach the subject from the perspective of memorising the formulae, and then memorising which kinds of questions demanded which kinds of formulae. We had the appearance of learning — we learnt the definitions of standard deviation, the normal distribution, the Poisson distribution, the expected value of a discrete random variable, and so forth, but I felt that not all of us were actually acquainted with these definitions.
Every time a trial exam or other test rolled around, I would encounter the same questions from my classmates: what's an example of a normal distribution? Why do you apply this principle instead of that? It reminds me in some way of Richard Feynman's experience teaching physics in Brazil, where he found that "Everything was entirely memorized, yet nothing had been translated into meaningful words." Most of my classmates know that an example of a normal distribution is the heights of the students in a class, and an example of a Poisson distribution is the number of cars that pass over a stretch of road during an hour, but I cannot tell how many can explain why, or give a different example besides the one we memorise.
In some part, the curriculum is to blame. The material we are exposed to emphasises a particular way of learning, which happens to be rote memorisation. We take notes, but all we do is jot down definitions in terms we do not necessarily completely understand. We learn that a discrete variable only takes non-integer values, but we can't explain why beyond regurgitating the definition we have been given.
The other thing to blame is our entire approach to teaching and learning altogether. From young, the mould we are expected to fit ourselves into emphasises rote learning. Nobody cares whether you understand, comprehend, grasp what you have been "taught" — what's important is that given the right question, you can give the right answer.
At the lower levels, it is rather easy to fake knowing something. But as you progress, it is harder to keep track of everything unless you can understand what you are doing — and, of course, it is difficult to come up with a new idea or principle when you don't understand the old ones. If all we care about is whether our students can factor a quadratic polynomial our education system is doing swell. But if we want our students to understand what they are taught so they can take it, apply it, and build on it, our education system is a dismal failure.