… this picture.
In a really great way Jan-Martin Klinge, who is a teacher of Mathematics, tries to explain to students and parents (unfortunately just in German), what learning mathematics is really all about. If you show this picture to a large group, regardless if they are students or parents – and they are not to communicate – there are very few who “get it” right away. The rest sees random black and white dots – maybe modern art? Same goes for maths – a selective few understand new theorems right when reading them.
For the rest Maths means: Work. It means to really look at this picture as long as it takes. You will eventually see it, and if you do, it’ll “click”. After that moment (which itself is in fact a pretty rewarding feeling), you will be 100% sure to know what this picture is all about, you’ll always see it, and no one will be able to confuse you.
If on the other hand, I told you, what you are supposed to see, you might just accept it, without really seeing it. Now you might know what you are supposed to see, but if someone else told you something different, you couldn’t verify it. And you couldn’t even reproduce it, e.g. by trying to explain someone else how to see it.
That’s in a digest, what the article is talking about, with the lesson to be learned being: It will take time and a lot of effort and discipline, until you understand something in Maths and there’s no way arround it. If you just accept it, you won’t get it, and in Maths where everything is consecutive to each other, you’ll never get anywhere, if you just accept.
The author speaks about Maths, but of course, this is a brilliant example that you can just transfer to any other subject as well. Take computer science, for example. You won’t ever get, what recursion is and how it works, if you never managed to think recursively. Try it on the “Tower of Hanoi“. If you understood recursion, you’ll always be able to find an elegant recursive solution, and anybody who understood recursion will get it. But if you didn’t, not a single thing of it will make any sense. And of course, you will never get programming, by just reading about it, and reproducing the examples. Only when you are forced to do something yourself will you really start to learn, to experience and thus – start to get it. Once you are there, it cannot be taken from you – but getting there requires a lot of time and effort. And in the beginning it will be pretty frustrating. The examples might be easy, but try developing your own ideas, after just learning the first basics of programming – you’ll probably fail with some error message that – at least in the beginning – doesn’t make any sense to you, you’ll try editing it, making it worse, you’ll search for explanations on the web, and in the end hours will have passed, untill you got rid of the error. And then you’ll realize that you haven’t even gotten 10% of your idea implemented. But believe me, you’ll take a lot out of this experience and in your second project everything will go much smoother and work like a charm.
Now, if you think carefully you might find yourself saying: “Hey, I study Physics” or “Hey I’m studying politics”, etc. “… and I can relate to that”. My thesis is, that even to be able to (critically) think, you need to work at it. And of course, regardless of what you do: where don’t you need to think? Someone who always just accepts and never questions something – regardless if it is the view of a fellow, an article in the news, or some decision a politician makes – will in the long run never be able to think for himself.
And to do so, you need time. To question something you need an opinion. To get to an – well founded – opinion, you need to know all the facts, and to get to there, you need to invest time and effort.
If you do, your reward will be, that you’ll see what’s in the picture. But of course, and that’s something that also the author of the article points out: Nowadays people don’t want or like to spend time and effort into Maths (and that is true for so many people in all the subjects that you might think of). But in the end it’s up to you – do you want to walk around being sure, knowing and seeing? Or will you end up walking this earth being blind?