Calculus by and for Young People

"This is a wonderful, very short book by a man called Don Cohen. It’s just an image, for me, of what’s possible. If we do turn the situation around and get children interested in exploring ideas and feeling confident enough to follow through on those ideas, then, actually, things that are said to be incredibly complex and you can’t do them till you’re 16, Don Cohen — as Gattegno did in his own work — has shown that to be nonsense. So Don Cohen gets children excited about ideas of infinity. One of the activities he does with very young children is getting them to consider what happens if you add 1/2 to a 1/4 and then add on an 1/8 and 1/16 and a 1/32 and so on to infinity. He does this by getting people to imagine a square. Picture a square in your mind and imagine shading half of it. Then shade a quarter — a half of the bit that’s unshaded. Then you shade an eighth – again, a half of what’s unshaded. And then you shade a sixteenth, again you’re shaving a half of what’s unshaded. Quite quickly, even young children can get a sense that they can carry on shading half of what they’re left with forever. They’re filling up more and more of this square, but there’s always going to be a tiny bit left over. They get a sense that there might be this infinite process. Those are extraordinary ideas, I think. On one level quite counter-intuitive, but incredibly exciting and certainly I’ve worked with children of 9,10,11 on them, and there’s no way they can’t get their head around them. In this little book that’s developed and he does go into calculus, he gets children finding areas under straight lines and generalizing. It’s an exciting book, and for me just an extraordinary image of what might be possible in the classroom."