Stage 1.

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.

Stage 2.

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.

Stage 3.

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 be identified as the mathematical content of those games that are similar to each other in structure, even though they might be totally different from the point of view of the elements used for playing them. It is even desirable, at one point, to establish “dictionaries” between games that have the same structure, so to each element and to each operation in one game, should correspond a unique element or operation in the other game. This will encourage learners to realize that the external material used for playing the games is less important than the rule structure which each material embodies. So learners will be encouraged to take the first halting steps towards abstraction, which is of course becoming aware of that which is common to all the games with the same rule structure, while the actual physical “playthings” can gradually become “noise”. This stage could be called the comparison stage.

Stage 4.

There comes a time when the learner has identified the abstract content of a number of different games and is practically crying out for some sort of picture by means of which to represent that which has been gleaned as the common core of the various activities. At this point it is time to suggest some diagrammatic representation such as an arrow diagram, table, a coordinate system or any other vehicle which would help fix in the learner’s mind what this common core is. We cannot ever hope to see an abstraction, as such things do not exist in the real world of objects and events, but we can invent a representation which would in some succinct way give the learner a snapshot of the essence that he has extracted or abstracted through the various game activities. Each one of the learned games can then be “mapped” on to this representation, which will pinpoint the communality of the games. This stage can be called the representation stage.

Stage 5.

It will now be possible to study the representation or “map” and glean some properties that all the games naturally must have. For example it could be checked whether a certain series of operations yields the same result as another series of operations. Such a “discovery” could then be checked by playing it out in one or more of the games whose representation yielded the “discovery”. An elementary language can then be developed to described such properties of the map. Such a language can approximate to the conventional symbolic language conventionally used by mathematicians or freedom can be exercised in inventing quite new and different symbol systems. Be it one way or another, a symbol system can now be developed which can be used to describe the properties of the system being learned, as the information is gathered by studying the map. This stage can be called the symbolization stage.

Stage 6.

The descriptions of the symbolization stage can get very lengthy and often quite redundant. There comes a time when it becomes desirable to establish some order in the maze of descriptions. This is the time to suggest that possibly just a few initial descriptions would suffice, as long as we appended ways of deducing other properties of the map, determining certain definite rules that would be allowed to be used in such “deductions”. In such a case we are making the first steps towards realizing that the first few descriptions can be our AXIOMS, and the other properties that we have deduced can be our THEOREMS, the ways of getting from the initial axioms to the theorems being the PROOFS. This stage could be called the formalization stage.