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PIAGET'S VIEWPOINT ON THE TEACHING OF PROBABILITY: A BREAKING-OFF WITH THE TRADITIONAL NOTION OF CHANCE?

Piaget’s constructivism and its further developments are used as the conceptual framework to relate, in the learning process, students’ age with specific topics in probability and statistics. Such a perspective consists of opposing the notion of chance to that of a reversible sequence and, therefore, to causality. Nevertheless, when the contributions to probability theory developed during the 17 th to the 19 th centuries are considered, it can be noticed that the concept of chance is a characterization of “our ignorance of the causal chain.” This fact motivates two questions which are discussed in this manuscript. The first one consists of understanding what constitutes the breaking-off between Cournot’s viewpoint of probability and the traditional one. The second question consists of exploring what kind of probability and statistics teaching would be developed if the traditional viewpoint on chance and probability is considered. FROM CAUSALITY TO THE NOTION OF CHANCE: PIAGET’S PROGRAM According to Piaget (1950) and Piaget and Inhelder (1974), children do not have an inborn notion of chance. It is acquired after the acquisition of the corresponding opposite notions. This means that chance events can be grasped only after identifying a set of phenomena which are not random events. Probability is a secondary notion in the sense that it is a synthesis of two concepts, namely chance and operations: it allows us to provide a structure to the field of fortuitous dispersions. Notions opposite to chance are characterized in terms of the basic structures acquired during the mental development of children. According to Piaget, this development has a central feature: to transit from reversible and composable operations to the irreversible ones. The analysis of the individual evolution of intelligence addresses the following problem: how the acquisition of logical and mathematical operations depends on concrete actions which in turn are reversible and composable? This development is characterized by three different stages. First stage (before 7-8 years old): this period is characterized by the absence of reversible compositions, of nested hierarchies and of totality conservation. At this stage, children can only articulate intuitive relationships. Second stage (between 7-8 and 11-12 years old): this period is characterized by the construction of logical operations when clustering and/or counting concrete objects (i.e., objects that can be directly handled). Third stage (from 11-12 years old): this period is characterized by formal thinking, namely the ability to link different concrete operational systems applied to different concrete objects, and to translate them in terms of logical implications (propositional logic). As explained in Beth and Piaget (1961), the structures empirically observed during the mental development of children are related with the irreducible mathematical structures in Bourbaki’s sense, namely that a mathematical structure is characterized by its syntax (i.e., a set of rules for combining elements) and that the syntactic properties do not depend on semantic considerations (i.e., concrete meanings of rules and elements). According to Piaget, three types of irreducible structures are empirically observed and so related with specific mathematical structures: firstly, structures such that reversibility corresponds to inversion or cancellation - these ones are similar to algebraic models such groups; second, structures such that the reversibility is the reciprocity - these ones are similar to order structures such lattices; third, structures based on elementary topological homomorphism - these ones are related to spatial structures. Notions opposite to chance are, therefore, based on systems of reversible operations which can be composed between them, namely logical and arithmetical operations. Reversibility and composability allow us to carry out clustering and build groups of objects. However, it can be observed that there are events irreducible to these structures. This fact leads Piaget to consider chance as opposed to two types of causality: firstly, chance is distinct from the mechanistic determinism in which the spatial-temporal connections are