Let me ramble away a bit on thie petunia problem, because I think it gives an insight between the difference between what mathematics is and what schools call maths but is actually either arithmetic or perhaps a little beyond that. The "arithmetic" way of solving the problem is to perhaps draw a little sketch, see that there are four corners, and four strips each one foot wide and the length of one of sides, work out all the bits and add them together. That works, but, as mathematicians like to say, lacks generality. Suppose we had the same situation, but instead of it being a nice one foot border all of the way round, we rotated the inner rectangle so the centres of the larger and smaller rectangles coincided, but the corners of the inner touched the sides of the outer. Now the person who is following the arithmetic all approach schools teach finds themselves in a nightmare set of calculations: where exactly to the touch? What is the angle? What is the area of each of the triangles? .. And so on, all of which is is unnecessary, since all that is needed is to to observe that the area of the petunias is the total area minus the area that is not petunia, and position, orientation, where lines meet and all the rest is of no importance.

And that's why I don't think many schools teach mathematics.


And as a further generalisation in honour of pi day let me express Pythagoras theorem in an unconventional way: the circle on the hypotenuse is equal to the sum of the circles on the other two sides.