Using a picture, I like to explain to my students (and parents) how mathematics works. I then ask them to look at the following picture in silence, without telling the others what they think they see in it.
Most students (and everyone else) see nothing but a wild pattern of black and white spots on the picture. Modern art or something.
But two or three of them understand the pattern. You see what is shown here.
And the same goes for mathematics. You always have one or two talented people who – apparently without learning – go straight through. The ones write only ones and easily familiarize themselves with each topic.
But the big rest doesn’t understand anything at first. Black and white spots. a2+b2 = …what?
I then give the pupils time. Show them the picture again and again. And then a „Aahhh!“ comes from here, a „Now I see it“ from there. Most students need time.
Much
Time.
With the picture as well as with the mathematics. And the longer it takes them to recognize what is to be seen here, the more intensively they experience this moment of „Aahhh“, this moment of understanding. When a confused pattern suddenly turns into a meaningful picture. A photo. „Now I see it too!
The problem with mathematics is that many students don’t feel like working out the content. Because that is exhausting and costs time. They prefer to be explained what can be seen on the picture. But if you don’t understand it yourself, then you don’t have this moment of understanding. This moment when a switch in your head tips over.
For this reason I am a consistent opponent of permanent coaching in the school system. To really understand – in such a way that one never forgets it – is only possible if one sits down and works it out for oneself. In my lessons everything is always explained at least twice. Every rule, every calculation is shown step by step in the book. Permanent tutoring releases the students from thinking for themselves. And I believe that most students will get worse in the long run.
What is in the picture?
Well, I won’t tell you that! But if you recognize it, then you are absolutely sure. Then there’s no doubt that you know what’s in the picture. So if you are not 100% convinced… then you have not yet recognized it.