The maze

Solving a maths problem is like working one's way through maze. It is exploratory. In maths one has to rearrange the expressions constantly, simplifying them, trying to spot something one has seen before that might give a clue to the route thereafter. Many pupils of mathematics do not share this conceptual model of a maths problem. To them maths consists of the statement of a problem to which they have to match up the routine which will solve that problem. This is a huge burden for the pupil. They have to remember so many routines, and they have to be able to remember the occasions on which they are to be used. As they progress through their education the mathematical problems they encounter become more and more complex and there is no single routine that can be applied to provide a solution. At this point the pupil finds that they cannot do maths anymore. This method of doing maths probles is a much more difficult way than that encountered by the pupil who imagines the maths problems from the point of view of the maze walker. All this latter pupil has to do is simply rearrange whatever expression he has in front of him over and over again, simplifying it and trying to shape it into something that is able to produce a solution. On occasion the maze walking pupil will change the expression in a way that is unhelpful and have to stop and go back to a point where he is confident that he is at a reasonably sound position and then rearrange the mathematical expression differently to take another path. The problems he has to solve are the smaller immediate problems of what rearrangement to do next. He can ignore the bigger picture because he knows that as long as he obeys the rules of maths whatever expression he comes up with will be valid (whether or not it is helpful in producing a solution).
This trial and error approach makes maths easier, but it requires sufficient confidence on the part of the pupil to make mistakes, and then go on and correct them. At Puppet Maths we teach pupils to treat maths problems as a maze or a jungle through which they have to find their path, by skill, by guile or even through inspiration.