**The theory of the six stages of learning with integers**

*(Published in “Mathematics in Schools”, Volume 29, Number 2, March 2000)*

**Stage 1 Free interaction**

This first stage will already have been experienced in children’s daily activities, such as siblings arguing with each other about one of them having more or less candy than the other, or going to bed later or earlier than the other or through any such simple “more or less” situation which we all encounter very early in life.

**Stage 2. Playing by the rules**

This stage, *“learning to play a game with rules”*, is the stage in which children discover some rules and learn to play by them . Alternatively own rules can be invested and play by them, as shown in the 3 examples.

**Stage 3. The comparison stage**

The stage of comparing ‘elements’ (objects) in the game.

Stage 4. Representation

This stage introduces mathematical representation. For example, instead of “more” we could use its Latin equivalent, namely the word “plus”. Also instead of “less” we could use the Latin word “minus”, which means “less”

**Stage 5. Symbolization**

The stage of “mathematical sentence”.

**Stage 6. Formalization**

Simplifying mathematical formulas to truely understand and appreciate the fine detail of the “proofs”.