Pathways to Number: Children's Developing Numerical Abilities

By Jacqueline Bideaud; Claire Meljac et al. | Go to book overview

11
The Appropriation of the Concept of Number: A Lengthy Process

Gerard Vergnaud LaPsydee

The study of the concept of number furnishes an excellent illustration of several key notions in the psychology of higher cognitive functions. It shows that the process of appropriation and construction of knowledge is a lengthy one, that the wide range of properties associated with a given concept are extremely diversified, and that acquisition of these properties is extremely slow. On a theoretical level, investigation of the concept of number points clearly to the impossibility of isolating the study of development from the study of learning, and the need to tie cognitive psychology and the epistemology of emergent knowledge and ability together.

By epistemology I am not referring to a general theory of knowledge but rather, on a more modest level, to recognition of the role of a given piece of specific knowledge--the relationship of given concepts and abilities--to the practical and theoretical problems that the solution of these problems is based on.

Although the most elementary aspects of the concept of number begin to be grasped by the age of 3 or 4, most individuals do not have a full understanding of signed (i.e., positive and negative) numbers and rational numbers before the age of 15 and they have a long way to go before they understand all the properties of real numbers, complex numbers, and the many other categories of numbers mathematicians have developed throughout history.

Few psychologists have taken an empirical interest in the learning of mathematics: The general feeling is that it is not of direct concern to psychologists. However, very young children--both in school and out--encounter situations for which analysis employing the concepts of signed, rational, and real numbers is relevant. In addition, children soon develop fairly good intuitions about these situations: For example, a number of studies in education show that by the age of 9 children are able to understand the concept of unlimited approximation by a

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