Learn, combine, repeat (a thousand times) pt.2

(Continued from yesterday)

I think it’s better to have a solid base than to have broad superficial knowledge. Most high schools start repeating the material needed for central s in their last year or six months before the examinations. But that material mostly concentrates around the last subjects that have been offered. That means a command of basics is implicated, but where are those basics? Sometimes it’s good to turn back a bit further. I encounter kids in senior high school unable to tell why a^\log b^c=c\cdot a^log b. And in a way I don’t blame them. Years ago they were taught the intricacies, after that it was supposed to stay with them forever. As we all know repetition is the best way to learn and internalize knowledge, failing to repeat is often feailing to remember in the long run.

Methods and materials have changed since the years I left high school and university. I know. But the way that humans learn has not changed. Luckily there are a lot more possibilities to offer material for repetition. Think of video lessons on YouTube or initiatives like Khan Academy. Think of apps and programs that can generate literally thousands of different exercises.

But don’ forget all those elementary proofs we all learned. These are also valuable training possibilities to rehash old knowledge. Such exercises also help reintegrating old basic knowledge in new material, leading not only to a better understanding, but to better memorisation as well. And proofs are no ‘stupid’ exercises, but great ways to give you that cool nerdy feeling; the warm feeling that you achieved something. Concluding that an exercise gives you \frac{x^2}{x-1} as an answer might be okay, but showing why \sin(h+i)=\sin(h)\cos(i)+\sin(i)\cos(h) feels so much better.

The first solution tells you what math can achieve, the second tells you how beautiful it is. Like Hannibal in the ’80s series The A-Team: “I love it when a plan comes together.”