(Elaborated from Nesher, P. (1989). Microworlds in Mathematical Education: A Pedagogical Realism. In L.B. Resnick (Ed.), Knowing Learning and Instruction (pp. 187-215). Hillsdale, NJ: Lawrence Erlbaum Associates)
 
A. Provision of the mathematical content
Mathematics - is an abstract area, which is unfamiliar to the child: He is not familiar with the objects and relations, which are abstract mathematical concepts, and is not familiar with the formal language of mathematics. Moreover, the child's thinking, at this stage, as Piaget has noted, is still concrete.
 
One of the ways to cope with these difficulties, is to find a concrete exemplification world, which can model the mathematical world, and in which the child can operate, and understand the mathematical concepts and the language describing them.
 
The concrete system, which exemplifies the knowledge domain with all its elements, is called the exemplification domain, and has the following characteristics:
  1. It is possible to exemplify in this system, all of the objects, operations, and relations included in the knowledge domain that we want to teach.
  2. The objects in this system are familiar to the child from previous activities, and there is a language familiar to the child, called the model's language - which enables to define different operations and different relations among these objects.
  3. The child is able to reconstruct in his imagination, the activity with the objects, after he stops operating on the concrete objects themselves.
In order for the first characteristic to exist, there should be a one-to-one correspondence between the objects and the relations in the knowledge domain and the formal language describing them, on one hand, and the objects and the relations in the exemplification domain and the model language describing them, on the other.



According to this approach, there are three stages in the learning process:
First Stage - The child operates within a system of objects which is familiar to him (the model), and with a language which is familiar to him (the model's language), in order to learn new concepts and relations which exist in mathematics and are exemplified by the model;
Second Stage- The child performs with the model, activities by which he learns the mathematical language describing the relations among the objects of the model;
Third Stage- The child works within the knowledge domain, without using the objects of the model.
All three stages are connected to each other and are insaparatable. Even when the child is in the third stage and operates with the mathematical language, the activity is based on previous work with the objects and the model's language. When a question arises, relating to the mathematical objects, the child can go back to the model, in order to check it. From this point of view the concrete model serves as a feedback mechanism.
 
B. The use of mathematical knowledge in everyday life
 
This is the place to emphasize, that the use of structured exemplification objects, does not mean a lack of dealing with the child's "natural" environment. After the child knows the mathematical concepts, which he learned by the structured exemplification, there comes the stage of applying the acquired mathematical knowledge, to solving applied problems. At this point he looks at his natural surroundings with already acquired mathematical tools.
For example: On the first stage, the child learns the additive relation among the numbers 2, 3, and 5 (2+3=5, or 5-3=2, or 5-2=3), by using the Cuisenaire rods, which are the exemplification domain we chose for the teaching of addition and subtraction. After he knows well this relation, he arrives at solving a problem. For example: There were flowers in the jar. Two flowers dried out in the morning, and three dried out in the evening. How many flowers dried during the day?
Knowledge of the mathematical fact 2+3=5 enables the child to solve the problem, and to know that 5 flowers dried out, without counting actual flowers. This is after the child made sure that the addition operation is indeed appropriate for the solution of the situation described in the question. The decision about what is the appropriate mathematical structure for the described situation, is the main task at this stage of solving situations from the natural environment.
 
The learning process may be summarized as follows: