Most people, when confronted with a situation which they are not sure how to handle, will engage in what is usually described as “trial and error” activity. What they are doing is to freely interact with the situation presented to them. In trying to solve a puzzle, most people will randomly try this and that and the other until some form of regularity in the situation begins to emerge, after which a more systematic problem solving behaviour becomes possible. This stage is the FREE PLAY, which is or should be, the beginning of all learning. This is how the would-be learner becomes familiar with the situation with which he or she is confronted.
After some free experimenting, it usually happens that regularities appear in the situation, which can be formulated as “rules of a game”. Once it is realized that interesting activities can be brought into play by means of rules, it is a small step towards inventing the rules in order to create a “game”. Every game has some rules, which need to be observed in order to pass from a starting state of things to the end of the game, which is determined by certain conditions being satisfied. It is an extremely useful educational “trick” to invent games with rules which match the rules that are inherent in some piece of mathematics which the educator wishes the learners to learn. This can be or should be the essential aspect of this part of the learning cycle. We could call this stage learning to play by the rules, as opposed to the free learning characteristic of stage one.
Once we have got children to play a number of mathematical games, there comes a moment when these games can be discussed, compared with each other. It is good to teach several games with very similar rule structures, but using different materials, so that it should become apparent that there is a common core to a number of different looking games, which can later...