A picture paints a thousand words

Mathematics is about applying logical reasoning to solve problems. But logical reasoning can be hard work. No one likes hard work. That's why tasks that have been thought out already are avoided and the known solutions applied. However, just memorising and applying these known solutions is not interesting or stimulating, nor does it teach the child to tackle new types of problem. Puppet Maths encourages pupils to use logical reasoning when approaching a maths problem. So how do we avoid the hard work? We do it by teaching pupils to visualise the problem. Rearranging counters into patterns and recognising these patterns is so much easier than counting; creating a graphical representation is so much easier than reading a table of figures (which leads to the question of why company accounts are presented as a table of figures rather than in some diagramatic format, are they being made deliberately difficult to interpret?).
When pupils learn to visualise the problems and the routines that they are using to solve their maths problems, they are learning to apply logic. This stands them in good stead for when a problem is presented to them that is in a different format from those which they are used to. It is this ability to apply logic that represents the real skill of mathematics. It is this skill that is so often not taught, or taught badly in schools.