Learning without knowing

I was reading Bert's blog yesterday and a question in his post stuck in my head for 20 hours. (But unfortunately, the original post was in Chinese so most of you will not be able to read it and since blogspot is blocked in China, my Chinese-knowing friends won't be able to read my post...So, only our HK friends are left to read both posts.)

So Bert said when he was studying with HK kids, he discovered that they are very good at "using" the knowledge without understanding. In other words, there's a formula or something and Bert asks the question how we should look at the situation that students just use them without understanding why.

Another professor was talking to me about almost the same topic another day regarding a UCSD knot theory course. Traditionally, you are supposed to lay much mathematical foundation to the material and then teach the theory. Or that's what they did at Princeton. But here, another excellent professor (who also found me a lift to AMS meeting) somehow managed to teach the course "hand wavy" without getting much into the details.

First of all, if you want to survive or even thrive school, in my observable range, it is absolutely fine just being able to use anything they teach you. And the tests at HKU and UCSD are mostly brain-dead versions of how to use facts (with some UCSD graduate courses as exceptions). It may be a sad fact but it can only be the way it is especially for a math test. For one, most theorems etc. are too hard to reproduce in a test's time. It took some great mathematicians at least several years to prove them or even a lot of mathematicians hundreds of years to prove them so they are highly non-trivial. And we are not all that great. For another, asking to reproduce a proof is unfair in a way (but tests may never even be fair...) because that's more like a reciting competition rather than something more meaningful. So in the trade-off between theory and application, I guess it's more sensible to choose application over theory.

But is it good? I mean being able to use them is definitely better than not being able to use them. However, there are harms being able to only use them. Generally, when the circumstances change or you need to somehow change or develop something new, dabblers fail. If all people only know known conclusions, there will hardly be new knowledge. And the biggest discoveries are always fundamental, like all things fall at the same speed (with no air resistance etc.).

I always want to know why and knowing only the fact never satisfies me. It's not practical or good all times but I believe knowing why stands for being able to create and knowing only how stands for being able to duplicate.