Kamii, C., & DeVries, R. (1976). Piaget, Children, and Number: Applying Piaget’s theory to the Teaching of Elementary Number. Washington, D.C: National Association for the Education of Young Children. ISBN: 0-912674-49-0
(Guest Post by Alprata Ahuja)
An Overview (pp. 1-4)
The question of how to apply Piaget’s theory to teaching of elementary number in classrooms is explored in this book. If we talk of direct application such as tasks where one to one correspondence (bridge) is established, concretely or through icons, between each element of one set to corresponding element of the other set than this does not lead to logico mathematical understanding of nature of number. This in consequence leads to a query in mind – then how should numbers be taught to children? In what form, Piaget’s theory can be applied to classrooms?
The Nature of Number (pp. 5-10)
To understand how number is an example of logico mathematical knowledge, we first look at the distinction between logico mathematical and physical knowledge. To understand further, the authors also draw out differences between logico mathematical knowledge and social knowledge.
“Physical knowledge is knowledge about objects in external reality” (p. 5). Through our observation, we can abstract physical properties of objects. Simple or empirical abstraction is the term used to describe this process. In this process of abstraction, child focuses on a particular physical property of the object and ignores the rest.
Logico mathematical knowledge is formed when relationship is constructed mentally between objects like “longer than”, “as heavy as”, “difference in colour” (p.5). This property does not exist in one object or another but in fact is the mental relationship constructed between objects. Numberness like twoness is also example of logico mathematical knowledge. The process of reflective or constructive abstraction is used to build logico mathematical relationship of objects. “This abstraction is a veritable construction by the mind rather than a focus on properties that already exist in objects” (p. 6).
Interestingly, proponents of “Modern Math” (p. 6) see understanding of numberness as equivalent to understanding number property common across different sets such as “four pencils, four flowers, four balloons” etc. (p. 6), which is against the logico mathematical structure of number as suggested by Piaget. This idea of Piaget becomes more significant and relevant to the readers in abstraction of larger numbers such as “100002” (p. 7), where construction of sets of 100002 objects is not as easy as construction of smaller sets.
“Social Knowledge is knowledge built by social consensus” (p. 8). It is formed by social transmission and has element of arbitrariness to it. Numbers can not be taught as social knowledge. “In fact, every culture that builds any mathematics at all ends up building exactly the same mathematics, as mathematics is built on the internal consistency of a deductive system in which absolutely nothing is arbitrary” (p. 8). No amount of drill will help child to understand number.
According to Piaget, number is synthesis of two kinds of relationships – Ordering and Class Inclusion. Ordering helps the child to build mental order to make sure that any object is not left or counted more than once. Also, for child to understand number as quantification of group of objects, child has to understand the relation of class inclusion for numbers, such as child should include “one in two, two in three, three in four” etc (p. 8). “A set of 8 objects will be quantified if he or she puts all 8 of them into a single relationship” (p. 8).
In the next chapter, what role teachers can play in teaching of numbers is explored as “principles of teaching that we devise from Piaget’s theory of number” (p. 10).